Manipulation of objects in potential energy landscapes

ABSTRACT

Holographic optical traps using the forces exerted by computer-generated holograms to trap, move and otherwise transform mesoscopically textured materials. The efficacy of the present invention is based upon the quality and nature of the diffractive optical element used to create the traps and dynamically use them. Further a landscape of potential energy sites can be created and used to manipulate, sort and process objects.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application is a divisional application and claims priority to U.S.patent application Ser. No. 11/285,224 filed on Nov. 22, 2005, whichclaims priority to U.S. Provisional Patent Application No. 60/630,378filed on Nov. 23, 2004; U.S. Provisional Patent Application No.60/656,482 filed on Feb. 24, 2005; and U.S. Provisional PatentApplication No. 60/663,218 filed on Mar. 18, 2005 and these applicationsare incorporated herein by reference in their entirety.

This work was supported by the National Science Foundation through GrantNumber DBI-0233971, with additional support from Grant NumberDMR-0451589.

FIELD OF THE INVENTION

The present invention relates generally to the creation and use ofholographic optical traps. More particularly, the present inventionrelates to the optimization of three-dimensional configurations ofholographic optical traps, to the creation of multiple state thermalratchets for media locomotion and manipulation, and to the creation anduse of potential energy well/peak landscapes for fractionation andsorting functionalities.

BACKGROUND OF THE INVENTION

A single laser beam brought to a focus with a strongly converging lensforms a type of optical trap widely known as an optical tweezer.Multiple beams of light passing simultaneously through the lens' inputpupil yield multiple optical tweezers, each at a location determined byits beam's angle of incidence and degree of collimation at the inputpupil. The trap-forming laser beams form an interference pattern as theypass through the input pupil, whose amplitude and phase corrugationscharacterize the downstream trapping pattern. Imposing the samemodulations on a single incident beam at the input pupil would yield thesame pattern of traps, but without the need to create and direct anumber of independent input beams. Such wavefront modification can beperformed by a type of diffractive optical element (DOE) commonly knownas a hologram.

Holographic optical trapping (HOT) uses methods of computer-generatedholography (CGH) to project arbitrary configurations of optical traps,with manifold applications in the physical and biological sciences, aswell as in industry. This flexible approach to manipulate and transformmesoscopic matter has been used to assemble two- and three-dimensionalstructures, to sort objects ranging in size from nanoclusters to livingcells, and to create all-optical microfluidic pumps and mixers.

SUMMARY OF THE INVENTION

The present invention involves refinements of the basic HOT techniquethat help to optimize the traps' performance, as well as a suite ofstatistically optimal analytical tools that are useful for rapidlycharacterizing the traps' performance. A number of modifications to theconventional HOT optical train minimize defects due to limitations ofpractical implementations. A direct search algorithm for HOT DOEcomputation can be used that is both faster and more accurate thancommonly used iterative refinement algorithms. A method for rapidlycharacterizing each trap in a holographic array is also described. Theoptimal statistical methods on which this characterization technique isbased lends itself naturally to digital video analysis of opticallytrapped spheres and can be exploited for real-time optimization.

In accordance with another aspect of the present invention theholographic traps are used to implement thermal ratchets in one, two,and three dimensions. A radial three-state ratchet is illustrated whereparticles of different size accumulate at different radial positions. Aradial two-state ratchet may be achieved as a two-dimensional extensionof the methods described above as well.

In accordance with yet another aspect of the present invention, theratchet is spherical. This is an extension of the two-dimensional radialratchet to three dimensions where spherical arrays of optical traps orother forms of potential energy wells sort and accumulate particles orobjects of different sizes at different spherical positions. This maytake the form of a two-state or three-state ratchet.

The two-state and three-state ratchets provide for a number of methodsand apparatus for locomotion. These methods of locomotion are achievedbroadly through the use of multiple potential energy wells. Thepotential energy wells may be achieved through a variety of methods. Inthe description above the method used is arrays of holographic opticaltraps. Additional methods include use of other photonic methods toimplement two-state and three-state ratchets based on the variousavailable methods of light steering. These also include chemical,biological, electrical, or other various mechanical methods involvingtwo-state or three-state ratchets in which an aspect of the presentinvention may include a movable lever or arm. This movable lever or armmay be user controllable so as to enable a number of devices ormechanisms which sort or pump or provide various forms of locomotion forparticles or objects.

In accordance with yet another aspect of the present invention is apolymer walker. This may be created through a walker shaped more or lesslike a capital Greek letter lambda out of a polymeric material such as agel that responds to an external stimulus by changing the opening anglebetween its legs. Examples of such active materials include Tanaka gels,which can be functionalized to respond to changes in salt concentration,electrolyte valence, pH, glucose concentration, temperature, and evenlight. These gels respond to such stimuli by swelling or shrinking. Thiscan be used to achieve the kind of motion described above for atwo-state ratchet. The rates of opening and closing can be set bychemical kinetics in such a system. The legs' affinity for specificplaces on a substrate can be determined by chemical, biochemical, orphysical patterning of a suitable substrate, with the ends of the legsappropriately functionalized to respond to those patterns.

In accordance with yet another aspect of the present invention two stateand three-state ratchets are used to build micromachines which mayconsist of mesoscopic motors based on synthetic macromolecules ormicroelectromechanical systems (MEMS).

These ratchet mechanisms may be used in the fabrication of devices orapparatus which pump, sort, shuttle or otherwise transport or manipulateparticles or cargo in a variety of patterns with application to a rangeof fields including but not limited to sorting systems, transportsystems, shuttle systems, sensor systems, reconnaissance systems,delivery systems, fabrication systems, purification systems, filtrationsystems, chemical processing, medical diagnostics and medicaltherapeutics.

In accordance with yet another aspect of the present invention,potential energy wells may be created using any of various methods ofcreating potential energy landscapes including without limitationelectrophoresis, dielectrophoresis, traveling wave dielectrophoresis,programmable dielectrophoresis, CMOS dieletrophoresis, optically inducedeletrophoretic methods, acoustic traps, and hydrodynamic flows as wellas other various such methods. These methods may be programmable. Thesemethods may further be programmed or constructed and controlled so as tocreate potential energy landscapes and potential energy wells thatimplement the various ratchet and fractionation constructs as presentedabove.

In accordance with yet another aspect of the present invention,potential energy landscapes may be created from a class of sourcesincluding optical intensity fields, optically guided dielectrophoresis,and any other technique including surface relief on a textured surface.Furthermore the present invention may be implemented as an opticallyguided dielectrophoresis implementation of optical fractionation, (whichmay be referred to as optically guided dielectrophoretic fractionation)as well as an optically guided dielectrophoretic implementation ofoptical ratchets (which may be referred to as optically guideddielectrophoretic ratchets).

These and other objects, advantages and features of the invention,together with the organization and manner of operation thereof, willbecome apparent from the following detailed description when taken inconjunction with the accompanying drawings, wherein like elements havelike numerals throughout the several drawings described below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1( a) is a simplified schematic of a conventional HOTimplementation; FIG. 1( b) shows a modification to the conventional HOTdesign of FIG. 1( a) that minimizes the central spot's influence andeffectively eliminates ghost traps by having the source laser beamconverge as it passes through the DOE; and FIG. 1( c) shows anothermodification to the conventional HOT design where a beam block is placedin the intermediate focal plane within the relay optics to spatiallyfilter the undiffracted portion of the beam;

FIG. 2 shows a three-dimensional multifunctional holographic opticaltrap array created with a single phase-only DOE computed with the directsearch algorithm, wherein the top DOE phase pattern includes whiteregions corresponding to a phase shift of 2π radians and black regionscorresponding to 0, and wherein the bottom projected optical trap arrayis shown at z=−10 μm, 0 μm and +10 μm from the focal plane of a 100×, NA1.4 objective lens, with the traps being spaced by 1.2 μm in the plane,and the 12 traps in the middle plane consisting of l=8 optical vortices;

FIG. 3 is a plot showing the performance metrics for the hologram ofFIG. 2 as a function of the number of accepted single-pixel changes,where data includes the DOE's overall diffraction efficiency as definedby Eq. (14), the projected patterns RMS error from Eq. (15), and itsuniformity, 1−u, where u is defined in Eq. (16); and

FIG. 4( a) is a design for 119 identical optical traps in atwo-dimensional quasiperiodic array; FIG. 4( b) shows the trappingpattern projected without optimizations using the adaptive-additivealgorithm; FIG. 4( c) shows the trapping pattern projected withoptimized optics and an adaptively corrected direct search algorithm;and FIG. 4( d) shows a bright-field image of colloidal silica spheres1.58 μm in diameter dispersed in water and organized in the optical traparray, where the scale bar indicates 10 μm.

FIG. 5( a) is a schematic representation of a holographic thermalratchet embodiment; FIG. 5( b) shows focused light from a typical HOTpattern; FIG. 5( c) shows an aqueous dispersion of colloidal trappedsilica spheres interacting with the HOTs; and FIG. 5( d) shows theratchet effect;

FIG. 6 shows flux induced by a two-state holographic optical ratchet;

FIG. 7( a) shows stochastic resonance in the two-state optical thermalratchet for τ/L=0.125; and FIG. 7( b) shows dependence on well depth forthe optimal angle rate τ/π=0.193 and duty cycle;

FIG. 8( a) shows flux reversed in a symmetric three-state opticalthermal ratchet as a function of cycle period fixed manifold separation;and FIG. 8( b) shows this flux reversal as a function of inter-manifoldseparation L for fixed cycle period T;

FIG. 9 shows calculated ratchet-induced drift velocity as a function ofcycle period for various inter-manifold separation L from adeterministic limit, L=6.55 to the stochastic limit L=130;

FIG. 10( a) shows fractionation in a radial optical thermal ratchet withpatterns of concentric circular manifolds with L=4.7 μm; FIG. 10( b) isa mixture of large and small particles interacting with a fixed trappingpattern; FIG. 10( c) is small particles being collected and large onesexcluded at L=6.9 μm and t=4.5 sec.; and FIG. 10( d) shows largeparticles concentrated at L=5.3 μm and t=4.5 sec. (the scale bar is 10μm).

FIG. 11( a)-11(d) shows a four-part sequence of spatially symmetricthree-state ratchet potentials;

FIG. 12( a) shows cross-over from deterministic optical peristalsis atL=6.5σ to thermal ratchet behavior with flux reversed at L-130 for athree-state cycle of Gaussian well potentials at βV₀=8.5, σ=0.53 μm andD=0.33 μm²/sec.; FIG. 12( b) is for evenly spaced values of L with theimage of 20×5 array of holographic optical traps at L₀=6.7 μm; FIG. 12(c) is an image of colloidal silica spheres 1.53 μm in diameterinteracting with the array; FIG. 12( d) shows rate dependence of theinduced drift velocity for fixed inter-trap separation, L₀; FIG. 12( e)shows separation dependence for fixed inter-state delay, T=2 sec.;

FIG. 13 shows one complete cycle of a spatially symmetric two-stateratchet potential comprised of discrete potential wells;

FIG. 14( a) shows a displacement function ƒ(t); and FIG. 14( b) shows anequivalent-ratchet driving force;

FIG. 15 shows steady-state drift velocity as a function of the relativedwell time, T₂/T₁, for βV₀=3.04, L=5.2 μm, σ=0.80 μm and various valuesof T/t (optimized at T/τ=0.193);

FIG. 16( a) shows an image of 5×20 array of holographic optical traps atL=5.2 μm; FIG. 16( b) shows a video micrograph of colloidal silicaspheres 1.53 μm in diameter trapped in the middle row of the array atthe start of an experimental run; FIGS. 16( c) and 16(d) show timeevolution of the measured probability density for finding particles atT₂=0.8 sec. and T₂=8.6 sec., respectively, with T₁ fixed at 3 sec.; FIG.16( e) shows time evolution of the particles' mean position calculatedfrom the distribution functions in FIGS. 16( c) and 16(d) (the slopes oflinear fits provide estimates for the induced drift velocity, which canbe compared with displacements calculated with Eq. (89) for βV₀=2.5, andσ=0.65); and FIG. 16( f) shows measured drift speed as a function ofrelative dwell time T₂/T₁, compared with predictions of Eq. (88); and

FIG. 17 shows a model of a diffusive molecular motor.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS A. OptimizedHolographic Optical Traps

FIGS. 1( a)-1(c) show a simplified schematic of a holographic opticaltweezer optical train before and after modification. FIG. 1( a) shows asimplified schematic of a conventional HOT implementation, where acollimated beam of light from a source laser is imprinted with a CGH andthereafter propagates as a superposition of independent beams, each withindividually specified wavefront characteristics. These beams arerelayed to the input aperture of a high-numerical-aperture lens,typically a microscope objective, which focuses them into optical trapsin the objective's focal plane. FIG. 1( a) shows the CGH being projectedby a transmissive DOE. The same principle applies to reflective DOE'swith the appropriate modification of the optical train. It should benoted that the same objective lens used to form the optical traps alsocan be used to create images of trapped objects. The associatedillumination and image-forming optics are omitted from FIGS. 1( a)-1(c)for clarity.

Practical holograms only diffract a portion of the incident light intointended modes and directions. A portion of the incident beam is notdiffracted at all, and the undiffracted portion typically forms anunwanted trap in the middle of the field of view. The undiffractedportion of the beam is depicted with dark shading in FIG. 1( a). This“central spot” has been removed in previous implementations by spatiallyfiltering the diffracted beam. Practical DOE's also tend to projectspurious “ghost” traps into symmetry-dictated positions within thesample. Spatially filtering a large number of ghost traps generally isnot practical, particularly in the case of dynamic holographic opticaltweezers whose traps move freely in three dimensions. Projectingholographic traps in the off-axis Fresnel geometry, rather than theFraunhofer geometry, automatically eliminates the central spot. However,this implementation limits the number of traps that can be projected andalso does not address the formation of ghost traps.

FIG. 1( b) shows one improvement to the basic HOT design that minimizesthe central spot's influence and effectively eliminates ghost traps. InFIG. 1( b), the source laser beam is converging as it passes through theDOE. As a result, the undiffracted central spot focuses upstream of theobjective's normal focal plane. The degree of collimation of eachdiffracted beam, and thus the axial position of the resulting trap, canbe adjusted by incorporating wavefront-shaping phase functions into thehologram's design, thereby returning the traps to the focal volume. Thisexpedient allows the central spot to be projected into the coverslipbounding a sample, rather than the sample itself, thereby ensuring thatthe undiffracted beam lacks both the intensity and the gradients neededto influence a sample's dynamics.

An additional consequence of the traps' displacement relative to theconverging beam's focal point is that ghost traps are projected to thefar side of this point and, therefore, out of the sample volumealtogether. This constitutes a substantial improvement for processes,such as optical fractionation, which use a precisely specified opticalpotential energy landscape to continuously sort mesoscopic objects.

Even though the undiffracted beam may not create an actual trap in thismodified optical train, it still can exert radiation pressure on theregion of the sample near the center of the field of view. This is aparticular issue for large arrays of optical traps that requiresubstantial power in the source beam. The undiffracted portion then canproduce a central spot far brighter than any individual trap.

Illuminating the DOE with a diverging beam further reduces theundiffracted beam's influence by projecting some of its light out of theoptical train. In a thick sample, however, this has the deleteriouseffect of projecting both the weakened central spot and the undiminishedghost traps into the sample.

These issues can be mitigated by placing a beam block as shown in FIG.1( c) in the intermediate focal plane within the relay optics tospatially filter the undiffracted portion of the beam. Because thetrap-forming beams come to a focus in a different plane, they are onlyslightly occluded by the beam block, even if they pass directly alongthe optical axis. The effect of this occlusion is minimal forconventional optical tweezers and can be compensated by appropriatelyincreasing their relative brightness. Therefore, the effect of anarrangement as described in FIG. 1( c) is the elimination of theundiffracted beam without substantially degrading the optical traps.

Holographic optical tweezers' efficacy is determined by the quality ofthe trap-forming DOE, which in turn reflects the performance of thealgorithms used in their computation. Conventional implementations haveapplied holograms calculated by a simple linear superposition of theinput fields. In such situations, the best results are obtained withrandom relative phases or with variations on the classicGerchberg-Saxton and Adaptive-Additive algorithms. Despite their generalefficacy, these algorithms yield traps whose relative intensities candiffer substantially from their design values and typically project asubstantial fraction of the input power into ghost traps. These problemscan become acute for complicated three-dimensional trapping patterns,particularly when the same hologram also is used as a mode converter toproject multifunctional arrays of optical traps.

A faster and more effective algorithm for HOT DOE calculation based ondirect search is generally as follows. The holograms used forholographic optical trapping typically operate only on the phase of theincident beam, and not its amplitude. Such phase-only holograms, alsoknown as kinoforms, are far more efficient than amplitude-modulatingholograms, which necessarily divert light away from the beam. Kinoformsalso are substantially easier to implement than fully complex holograms.General trapping patterns can be achieved with kinoforms despite theloss of information that might be encoded in amplitude modulationsbecause optical tweezers rely for their operation on intensity gradientsand not local phase variations. However, it is necessary to find apattern of phase shifts in the input plane that encodes the desiredintensity pattern in the focal volume.

According to scalar diffraction theory, the (complex) field E({rightarrow over (r)}) in the focal plane of a lens of focal length ƒ isrelated to the field, u({right arrow over (ρ)})exp(iφ({right arrow over(φ)})), in its input plane by a Fraunhofer transform,

$\begin{matrix}{{{E( \overset{arrow}{r} )} = {\int{{u( \overset{arrow}{\rho} )}{\exp ( {\; {\phi ( \overset{arrow}{\rho} )}} )}{\exp ( {{- }\frac{k{\overset{arrow}{r} \cdot \overset{arrow}{\rho}}}{2\; f}} )}{^{2}\rho}}}},} & (1)\end{matrix}$

where u({right arrow over (ρ)}) and φ({right arrow over (ρ)}) are thereal-valued amplitude and phase, respectively, of the field at position({right arrow over (ρ)}) in the input pupil, and k=2^(π/λ) is the wavenumber of light of wavelength λ.

If u ({right arrow over (ρ)}) is the amplitude profile of the inputlaser beam, then φ({right arrow over (ρ)}) is the kinoform encoding thepattern. Most practical DOEs, including those projected with SLMs,consist of an array {right arrow over (ρ)}_(j) of discrete phase pixels,each of which can impose any of P possible discrete phase shiftsφ_(j)ε{0, . . . , φ_(P−1)}. The field in the focal plane due to such anN-pixel DOE is, therefore,

$\begin{matrix}{{{E( \overset{arrow}{r} )} = {\sum\limits_{j = 1}^{N}\; {u_{j}{\exp ( {\; \phi_{j}} )}{T_{j}( \overset{arrow}{r} )}}}},} & (2)\end{matrix}$

where the transfer matrix describing the propagation of light from inputplane to output plane is

$\begin{matrix}{{T_{j}( \overset{arrow}{r} )} = {{\exp ( {{- }\frac{k{\overset{arrow}{r} \cdot {\overset{arrow}{\rho}}_{j}}}{2\; f}} )}.}} & (3)\end{matrix}$

Unlike more general holograms, the desired field in the output plane ofa holographic optical trapping system consists of M discrete brightspots located at {right arrow over (r)}_(m):

$\begin{matrix}{{{E( \overset{arrow}{r} )} = {\sum\limits_{m = 1}^{M}{E_{m}( \overset{arrow}{r} )}}},{with}} & (4) \\{{{E_{m}( \overset{arrow}{r} )} = {\alpha_{m}{\delta ( {\overset{arrow}{r} - {\overset{arrow}{r}}_{m}} )}{\exp ( {\; \xi_{m}} )}}},} & (5)\end{matrix}$

where α_(m) is the relative amplitude of the m-th trap, normalized by

${{\sum\limits_{m - 1}^{M}\; {a_{m}}^{2}} = 1},$

and ξ_(m) is its (arbitrary) phase. Here, δ({right arrow over (r)})represents the amplitude profile of the focused beam of light in thefocal plane, which may be treated heuristically as a two-dimensionalDirac delta function. The design challenge is to solve Eqs. (2), (3) and(4) for the set of phase shifts ξ_(m), yielding the desired amplitudesα_(m) at the correct locations {right arrow over (r)}_(m) given u_(j)and T_(j)({right arrow over (ρ)}).

The Gerchberg-Saxton algorithm and its generalizations, such as theadaptive-additive algorithm, iteratively solve both the forwardtransform described by Eqs. (5) and (6), and also its inverse, takingcare at each step to converge the calculated amplitudes at the outputplane to the design amplitudes and to replace the back-projectedamplitudes, u_(j) at the input plane with the laser's actual amplitudeprofile. Appropriately updating the calculated input and outputamplitudes at each cycle can cause the DOE phase φ_(j), to converge toan approximation to the ideal kinoform, with monotonic convergencepossible for some variants. The forward and inverse transforms mappingthe input and output planes to each other typically are performed byfast Fourier transform (FFT). Consequently, the output positions {rightarrow over (r)}_(m) also are explicitly quantized in units of theNyquist spatial frequency. The output field is calculated not only atthe intended positions of the traps, but also at the spaces betweenthem. This is useful because the iterative algorithm not only maximizesthe fraction of the input light diffracted into the desired locations,but also minimizes the intensity of stray light elsewhere.

FFT-based iterative algorithms have drawbacks for computingthree-dimensional arrays of optical tweezers, or mixtures of moregeneral types of traps. To see this, one notes how a beam-splitting DOEcan be generalized to include wave front-shaping capabilities.

A diverging or converging beam at the input aperture comes to a focusand forms a trap downstream or upstream of the focal plane,respectively. Its wave front at the input plane is characterized by theparabolic phase profile

$\begin{matrix}{{{\phi_{z}( {\overset{arrow}{\rho},z} )} = \frac{k\; \rho^{2}z}{f^{2}}},} & (6)\end{matrix}$

where z is the focal spot's displacement along the optical axis relativeto the lens' focal plane. This phase profile can be used to move anoptical trap relative to the focal plane even if the input beam iscollimated by appropriately augmenting the transfer matrix:

T _(j) ^(z)({right arrow over (r)})=T _(j)({right arrow over (r)})K _(j)^(z)({right arrow over (r)}),  (7)

where the displacement kernel is

K _(j) ^(z)({right arrow over (r)})=exp(iφ _(z)({right arrow over(ρ)}_(j) ,z)),  (8)

The result, T_(j) ^(z)({right arrow over (r)}), replaces T_(j)({rightarrow over (r)}) as the kernel of Eq. (2).

Similarly, a conventional TEM beam can be converted into a helical modeby imposing the phase profile

φ_(l)({right arrow over (ρ)})=lθ,  (9)

where θ is the azimuthal angle around the optical axis and l is anintegral winding number known as the topological charge. Suchcorkscrew-like beams focus to ring-like optical traps known as opticalvortices that can exert torques as well as forces. Thetopology-transforming kernel K_(j) ^(i)({right arrow over(r)})=exp(iφ_(l)({right arrow over (ρ)}_(j))) can be composed with thetransfer matrix in the same manner as the displacement-inducing {rightarrow over (r)}_(m).

A variety of comparable phase-based mode transformations are described,each with applications to single-beam optical trapping. All can beimplemented by augmenting the transfer matrix with an appropriatetransformation kernel. Moreover, different transformation operations canbe applied to each beam in a holographic trapping pattern independently,resulting in general three-dimensional configurations of diverse typesof optical traps.

Calculating the phase pattern φ_(j) encoding multifunctionalthree-dimensional optical trapping patterns requires only a slightelaboration of the algorithms used to solve Eq. (2) for two-dimensionalarrays of conventional optical tweezers. The primary requirement is tomeasure the actual intensity projected by φ_(j) into the m-th trap atits focus. If the associated diffraction-generated beam has anon-trivial wave front, then it need not create a bright spot at itsfocal point. On the other hand, if it is assumed that φ_(j) creates therequired type of beam for the m-th trap through a phase modulationdescribed by the transformation kernel K_(j,m)({right arrow over (r)}),then applying the inverse operator, K_(j,m) ⁻¹({right arrow over (r)})in Eq. (2) would restore the focal spot.

This principle was first applied to creating three dimensional traparrays in which separate translation kernels were used to project eachdesired optical tweezer back to the focal plane as an intermediate stepin each iterative refinement cycle. Computing the light projected intoeach plane of traps in this manner involves a separate Fourier transformfor the entire plane. In addition to its computational complexity, thisapproach also requires accounting for out-of-focus beams propagatingthrough each focal plane, or else suffers from inaccuracies due tofailure to account for this light.

A substantially more efficient approach involves computing the fieldonly at each intended trap location, as

$\begin{matrix}{{{E_{m}( {\overset{arrow}{r}}_{m} )} = {\sum\limits_{j = 1}^{N}\; {{K_{j,m}^{- 1}( {\overset{arrow}{r}}_{m} )}{T_{j}( {\overset{arrow}{r}}_{m} )}{\exp ( {{- }\; \phi_{j}} )}}}},} & (10)\end{matrix}$

and comparing the resulting amplitude α_(m)=|E_(m)| with the designvalue. Unlike the FFT-based approach, this per-trap algorithm does notdirectly optimize the field in the inter-trap region. Conversely, thereis no need to account for interplane propagation. If the values of α_(m)match the design values, then no light is left over to create ghosttraps.

Iteratively improving the input and output amplitudes by adjusting theDOE phases, φ_(j), involves back-transforming from each updated E_(m)using the forward transformation kernels, K_(j,m)({right arrow over(r)}_(m)) with one projection for each of the M traps. By contrast, theFFT-based approach involves one FFT for each wave front type within eachplane and may not converge if multiple wave front types are combinedwithin a given plane.

The per-trap calculation suffers from a number of shortcomings. The onlyadjustable parameters in Eqs. (5) and (10) are the relative phases ξ_(m)of the projected traps. These M−1 real-valued parameters must beadjusted to optimize the choice of discrete-valued phase shifts, φ_(j),subject to the constraint that the amplitude profile u_(j) matches theinput laser's. This problem is likely to be underspecified for bothsmall numbers of traps and for highly complex heterogeneous trappingpatterns. The result for such cases is likely to be opticallyinefficient holograms whose projected amplitudes differ from their idealvalues.

Equation (10) suggests an alternative approach for computing DOEfunctions for discrete HOT patterns. The operator, K_(j,m) ⁻¹({rightarrow over (r)}_(m))T_(m)({right arrow over (r)}_(m)) describes howlight in the mode of the m-th trap propagates from position {right arrowover (ρ)}_(j) on the DOE to the trap's projected position {right arrowover (r)}_(m), in the lens' focal plane. If the DOE's phase φ_(j) werechanged at that point, then the superposition of rays composing thefield at {right arrow over (r)}_(m) would be affected. Each trap wouldbe affected by this change through its own propagation equation. If thechanges led to an overall improvement, then one would be inclined tokeep the change, and seek other such improvements. If, instead, theresults were less beneficial, φ_(j) would be restored to its formervalue and the search for other improvements would continue. This is thebasis for direct search algorithms, including the extensive category ofsimulated annealing and genetic algorithms.

In its most basic form, direct search involves selecting a pixel atrandom from a trial phase pattern, changing its value to any of the P−1alternatives, and computing the effect on the projected field. Thisoperation can be performed efficiently by calculating only the changesat the M trap's positions due to the single changed phase pixel, ratherthan summing over all pixels. The updated trial amplitudes then arecompared with their design values and the proposed change is accepted ifthe overall amplitude error is reduced. The process is repeated untilthe acceptance rate for proposed changes becomes sufficiently small.

The key to a successful and efficient direct search for φ_(j) is toselect a function that effectively quantifies projection errors. Thestandard cost function,

∑_(m − 1)M(I_(m) − ɛ I_(m)^((D)))²,

assesses the mean-squared deviations of the m-th trap's projectedintensity I_(m)=|α_(m)|² from its design value I_(m) ^((D)), assuming anoverall diffraction efficiency of ε. It requires an accurate estimatefor ε and places no emphasis on uniformity in the projected traps'intensities. A conventional alternative,

C=−

l

+ƒσ,  (11)

avoids both shortcomings. Here,

${\langle I\rangle} = {\frac{1}{M}{\sum\limits_{m = 1}^{M}\; I_{m}}}$

is the mean intensity at the traps, and

$\begin{matrix}{\sigma = \sqrt{\frac{1}{M}{\sum\limits_{m = 1}^{M}\; ( {I_{m} - {\gamma \; I_{m}^{(D)}}} )^{2}}}} & (12)\end{matrix}$

measures the deviation from uniform convergence to the designintensities. Selecting

$\begin{matrix}{\gamma = \frac{\sum\limits_{m = 1}^{M}\; {I_{m}I_{m}^{(D)}}}{\sum\limits_{m = 1}^{M}\; ( I_{m}^{(D)} )^{2}}} & (13)\end{matrix}$

minimizes the total error and accounts for non-ideal diffractionefficiency. The weighting fraction ƒ sets the relative importanceattached to overall diffraction efficiency versus uniformity.

In the simplest direct search for an optimal phase distribution, anycandidate change that reduces C is accepted, and all others arerejected. Selecting pixels to change at random reduces the chances ofthe search becoming trapped by suboptimal configurations that happen tobe highly correlated. The typical number of trials required forpractical convergence should scale as N P, the product of the number ofphase pixels and the number of possible phase values. In practice, thisrough estimate is accurate if P and N are comparatively small. Forlarger values, however, convergence is attained far more rapidly, oftenwithin N trials, even for fairly complex trapping patterns. In thiscase, the full refinement requires computational resources comparable tothe initial superposition and is faster than typical iterativealgorithms by an order of magnitude or more.

FIG. 2 shows a typical application of the direct search algorithm tocomputing a HOT DOE consisting of 51 traps, including 12 opticalvortices of topological charge l=8, arrayed in three planes relative tothe focal plane. The 480×480 pixel phase pattern was refined from aninitially random superposition of fields in which amplitude variationswere simply ignored. The results in FIG. 2 were obtained with a singlepass through the array. The resulting traps, shown in the bottom threeimages, vary from their planned relative intensities by less than 5percent. This compares favorably with the 50 percent variation typicallyobtained with the generalized adaptive-additive algorithm. This effectwas achieved by setting the optical vortices' brightness to 15 timesthat of the conventional optical tweezers. This single hologramtherefore demonstrates independent control over three-dimensionalposition, wave front topology, and brightness of all the traps.

To demonstrate these phenomena more quantitatively, standard figures ofmerit are augmented with those known in the art. In particular, theDOE's theoretical diffraction efficiency is commonly defined as

$\begin{matrix}{{Q = {\frac{1}{M}{\sum\limits_{m = 1}^{M}\frac{I_{m}}{I_{m}^{(D)}}}}},} & (14)\end{matrix}$

and its root-mean-square (RMS) error as

$\begin{matrix}{e_{rms} = {\frac{\sigma}{\max ( I_{m} )}.}} & (15)\end{matrix}$

The resulting pattern's departure from uniformity is usefully gauged as

$\begin{matrix}{u = {\frac{\max ( {{I_{m}/I_{m}^{(d)}} - {\min ( {I_{m}/I_{m}^{(D)}} )}} )}{\max ( {{I_{m}/I_{m}^{(d)}} + {\min ( {I_{m}/I_{m}^{(D)}} )}} )}.}} & (16)\end{matrix}$

These performance metrics are plotted in FIG. 3 as a function of thenumber of accepted single-pixel changes. The overall acceptance rate forchanges after a single pass through the entire DOE array was better than16%.

FIG. 3 demonstrates that the direct search algorithm trades off a smallpercentage of the overall diffraction efficiency in favor ofsubstantially improved uniformity. This improvement over randomly phasedsuperposition requires little more than twice the computational time andtypically can be completed in a time comparable to the refresh intervalof a liquid crystal spatial light modulator.

Two-dimensional phase holograms contain precisely enough information toencode any two-dimensional intensity distribution. A three-dimensionalor multi-mode pattern, however, may require both the amplitude and thephase to be specified in the lens' focal plane. In such cases, atwo-dimensional phase hologram can provide at best an approximation tothe desired distribution of traps. Determining whether or not a singletwo-dimensional phase hologram can encode a particular trapping patternis remains an issue. The direct search algorithm presented above maybecome stuck in local minima of the cost function instead of identifyingthe best possible phase hologram. In such cases, more sophisticatednumerical search algorithms may be necessary.

The most straightforward elaboration of a direct search is the class ofsimulated annealing algorithms. Like direct search, simulated annealingrepeatedly attempts random changes to randomly selected pixels. Alsolike direct search, a candidate change is accepted if it would reducethe cost function. The probability P_(accept) for accepting a costlychange is set to fall of exponentially in the incremental cost ΔC,

$\begin{matrix}{P = {{\exp ( {- \frac{\Delta \; C}{C_{0}}} )}.}} & (17)\end{matrix}$

where C_(o) is a characteristic cost that plays the role of thetemperature in the standard Metropolis algorithm. Increasing C_(o)results in an increased acceptance rate of costly changes and adecreased chance of becoming trapped in a local minimum. The improvedopportunities for finding the globally optimal solution come at the costof increased convergence time.

The tradeoff between exhaustive and efficient searching can be optimizedby selecting an appropriate value of C_(o). However, the optimal choicemay be different for each application. Starting C_(o) at a large valuethat promotes exploration and then reducing it to a lower value that,speeds convergence offers one convenient compromise. Several strategiesfor varying C_(o) have been proposed and are conventionally recognized.

Substantially more effective searches may be implemented by attemptingto change multiple pixels simultaneously instead of one at a time.Different patterns of multi-pixel changes may be most effective foroptimizing trap-forming phase holograms of different types, and theapproaches used to identify and improve such patterns generally areknown as genetic algorithms. These more sophisticated algorithms may beapplicable for designing high-efficiency, high-accuracy DOEs forprecision HOT applications.

At least numerically, direct search algorithms are both faster andbetter at calculating trap-forming DOEs than iterative refinementalgorithms. The real test, however, is in the projected traps' abilityto trap particles. A variety of approaches have been developed forgauging the forces exerted by optical traps. The earliest approachinvolved measuring the hydrodynamic drag required to dislodge a trappedparticle, typically by translating the trap through quiescent fluid.This approach has several disadvantages, most notably that it identifiesonly the minimal escape force in a given direction and not the trap'sactual three-dimensional potential energy landscape. Mostconventionally-known implementations fail to collect sufficientstatistics to account for thermal fluctuations' role in the escapeprocess, and do not account adequately for hydrodynamic coupling tobounding surfaces.

Much more information can be obtained by measuring a particle'sthermally driven motions in the trap's potential well. One approachinvolves measuring the particle's displacements {right arrow over(r)}(t) from its equilibrium position and computing the probabilitydensity P({right arrow over (r)}) as a histogram of these displacements.The result is related to the potential energy well V({right arrow over(r)}) through the Boltzmann distribution

P({right arrow over (r)})=exp(−βV({right arrow over (r)})),  (18)

where β⁻¹=k_(B)T is the thermal energy scale at temperature T. Thermalcalibration offers benefits in that no external force has to be applied,and yet the trap can be fully characterized, provided enough data istaken. A complementary approach to thermal calibration involvescomputing the autocorrelation function of Δ{right arrow over (r)}(t).Both of these approaches require amassing enough data to characterizethe trapped particle's least probable displacements, and therefore mostof its behavior is oversampled. This does not present a significantissue when data from a single optical trap can be collected at highsampling rates, for example with a quadrant photodiode. Trackingmultiple particles in holographic optical traps, however, is mostreadily accomplished through digital video microscopy, which yields datamore slowly by two or three orders of magnitude. Fortunately, ananalysis based on optimal statistics provides all the benefits ofthermal calibration by rigorously avoiding oversampling.

In many cases, an optical trap may be modeled as a harmonic potentialenergy well,

$\begin{matrix}{{{V( \overset{arrow}{r} )} = {\frac{3}{2}{\sum\limits_{i = 1}^{3}{\kappa_{i}r_{i}^{2}}}}},} & (19)\end{matrix}$

with a different characteristic curvature K_(i) along each Cartesianaxis. This form has been found to accurately describe optical traps'potential energy wells in related studies and has the additional benefitof permitting a one-dimensional mathematical description. Consequently,the subscripts are dropped in the following discussion.

The behavior of a colloidal particles localized in a viscous fluid by anoptical trap is described by the Langevin equation

$\begin{matrix}{{{\overset{.}{x}(t)} = {{- \frac{x(t)}{\tau}} + {\xi (t)}}},} & (20)\end{matrix}$

where the autocorrelation time

$\begin{matrix}{\tau = \frac{\gamma}{\kappa}} & (21)\end{matrix}$

is set by the viscous drag coefficient γ and the curvature of the well,κ, and where ξ(t) describes random thermal forcing with zero mean,

ξ(t)

=0, and variance

$\begin{matrix}{{{\xi (t)}{\xi (s)}} = {\frac{2\; k_{B}T}{\gamma}{{\delta ( {t - s} )}.}}} & (22)\end{matrix}$

If the particle is at position x₀ ^(x) ⁰ at time t=0, its trajectory atlater times is given by

$\begin{matrix}{{x(t)} = {{x_{0}{\exp ( {- \frac{t}{\tau}} )}} + {\int_{0}^{t}{{\xi (s)}{\exp ( {- \frac{t - s}{\tau}} )}\ {{s}.}}}}} & (23)\end{matrix}$

Experimentally, such a trajectory is sampled at discrete timest_(j)=jΔt, so that Eq. (23) may be rewritten as

$\begin{matrix}{x_{j + 1} = {{{\exp ( {- \frac{t_{j - 1}}{\tau}} )}x_{0}{\int_{0}^{t_{j}}{{\xi (s)}{\exp ( {- \frac{t_{j} - s}{\tau}} )}\ {s}}}} + {\int_{t_{j}}^{t_{j + 1}}{{\xi (s)}{\exp ( {- \frac{t_{j + 1} - s}{\tau}} )}\ {s}}}}} & (24) \\{{= {{\varphi \; x_{j}} + a_{j + 1}}},{where}} & (25) \\{\varphi = {\exp ( {- \frac{\Delta \; t}{\tau}} )}} & (26)\end{matrix}$

and where α_(j+1) is a Gaussian random variable with zero mean andvariance

$\begin{matrix}{\sigma_{a}^{2} = {{\frac{k_{B}T}{\kappa}\lbrack {1 - {\exp ( {- \frac{2\; \Delta \; t}{\tau}} )}} \rbrack}.}} & (27)\end{matrix}$

Because φ<1, Eq. (25) is an example of an autoregressive process whichis readily invertible. In principle, the particle's trajectory {x_(j)}can be analyzed to extract φ and σ_(a) ², which, in turn, provideestimates for the trap's stiffness, κ, and the viscous drag coefficientγ.

In practice, however, the experimentally measured particle positionsy_(j) differ from the actual positions x_(j) by random errors b_(j),which is assumed to be taken from a Gaussian distribution with zero meanand variance σ_(b) ². The measurement then is described by the coupledequations

x _(j) =φx _(j−1) +a _(j) and

y _(j) =x _(j) +b _(j),  (28)

where b_(j) is independent of a_(j). Estimates can still be extractedfor φ and as from a set of measurements σ_(a) ² by first constructingthe joint probability

$\begin{matrix}{{p( {\{ x_{i} \},{\{ y_{i} \} \varphi},\sigma_{a}^{2},\sigma_{b}^{2}} )} = {\prod\limits_{j = 2}^{N}\; {\lbrack \frac{\exp ( {- \frac{a_{j}^{2}}{2\; \sigma_{a}^{2}}} )}{\sqrt{2\; \pi \; \sigma_{z}^{2}}} \rbrack {\prod\limits_{j = 1}^{N}\mspace{11mu} \lbrack \frac{\exp ( {- \frac{b_{j}^{2}}{2\; \sigma_{b}^{2}}} )}{\sqrt{2\; \pi \; \sigma_{b}^{2}}} \rbrack}}}} & (29) \\{= {\prod\limits_{j = 2}^{N}\; {\lbrack \frac{\exp ( {- \frac{( {x_{j} - {\varphi \; x_{j - 1}}} )^{2}}{2\; \sigma_{a}^{2}}} )}{\sqrt{2\; \pi \; \sigma_{z}^{2}}} \rbrack {\prod\limits_{j = 1}^{N}\mspace{11mu} {\lbrack \frac{\exp ( {- \frac{( {y_{j} - x_{j}} )^{2}}{2\; \sigma_{b}^{2}}} )}{\sqrt{2\; \pi \; \sigma_{b}^{2}}} \rbrack.}}}}} & (30)\end{matrix}$

The probability density for a given set of measurements is obtained byintegrating over all trajectories,

$\begin{matrix}\begin{matrix}{{p( {{\{ y_{j} \} \varphi},\sigma_{a}^{2},\sigma_{b}^{2}} )} = {\int{{p( {\{ x_{j} \},{\{ y_{j} \} \varphi},\sigma_{a}^{2},\sigma_{b}^{2}} )}{x_{1}}\mspace{11mu} \ldots \mspace{11mu} {x_{N}}}}} \\{{= {\frac{( {2\; \pi \; \sigma_{a}^{2}\sigma_{b}^{2}} )^{- \frac{N - 1}{2}}}{\sqrt{\sigma_{b}^{2}{\det ( A_{\varphi} )}}}{\exp ( {{- \frac{1}{2\; \sigma_{b}^{2}}}{( \overset{arrow}{y} )^{T}\lbrack {I - \frac{A_{\varphi}^{- 1}}{\sigma_{b}^{2}}} \rbrack}\overset{arrow}{y}} )}}},}\end{matrix} & (31)\end{matrix}$

where {right arrow over (y)}=(y₁ . . . y_(N)), ({right arrow over(y)})^(T) is its transpose, I is the N×N identity matrix, and

$\begin{matrix}{{A_{\varphi} = {\frac{I}{\sigma_{b}^{2}} + \frac{M_{\varphi}}{\sigma_{a}^{2}}}},} & (32)\end{matrix}$

with the memory tensor

$\begin{matrix}{M_{\varphi} = {\begin{pmatrix}\varphi^{2} & {- \varphi} & 0 & 0 & \ldots & 0 \\{- \varphi} & {1 + \varphi^{2}} & {- \varphi} & 0 & \ldots & \vdots \\0 & {- \varphi} & {1 + \varphi^{2}} & {- \varphi} & \ldots & \vdots \\0 & 0 & {- \varphi} & \ddots & \ldots & \vdots \\\vdots & \vdots & \ldots & {- \varphi} & {1 + \varphi^{2}} & {- \varphi} \\0 & 0 & \ldots & 0 & {- \varphi} & {- 1}\end{pmatrix}.}} & (33)\end{matrix}$

Calculating the determinant, det(A_(φ)), and inverse A_(φ) ⁻¹, isgreatly facilitated if time translation invariance is artificiallyimposed by replacing M_(φ) with the (N+1)×(N+1) matrix

$\begin{matrix}{{\hat{M}}_{\varphi} = {\begin{pmatrix}{1 + \varphi^{2}} & {- \varphi} & 0 & 0 & \ldots & {- \varphi} \\{- \varphi} & {1 + \varphi^{2}} & {- \varphi} & 0 & \ldots & \vdots \\0 & {- \varphi} & {1 + \varphi^{2}} & {- \varphi} & \ldots & \vdots \\0 & 0 & {- \varphi} & \ddots & \ldots & \vdots \\\vdots & \vdots & \ldots & {- \varphi} & {1 + \varphi^{2}} & {- \varphi} \\{- \varphi} & 0 & \ldots & 0 & {- \varphi} & {1 + \varphi^{2}}\end{pmatrix}.}} & (34)\end{matrix}$

Physically, this involves imparting an impulse, α_(N+1), that translatesthe particle from its last position, x_(N), to its first, x₁. Becausediffusion in a potential well is a stationary process, the effect ofthis change decays as 1/N in the number of measurements, and so is lessimportant than other sources of error.

With this approximation, the determinant and inverse of A_(φ) are givenby

$\begin{matrix}{{{\det ( A_{\varphi} )} = {\prod\limits_{n = 1}^{N}\; \{ {\frac{1}{\sigma_{b}^{2}} + {\frac{1}{\sigma_{a}^{2}}\lbrack {1 + \varphi^{2} - {2\; \varphi \; {\cos ( \frac{2\; \pi \; n}{N} )}}} \rbrack}} \}}}{and}} & (35) \\{{{( A_{\varphi}^{- 1} )\alpha \; \beta} = {\frac{1}{N}{\sum\limits_{n = 1}^{N}\frac{\sigma_{a}^{2}\sigma_{b}^{2}{\exp ( {\frac{2\; \pi}{N}{n( {\alpha - \beta} )}} )}}{\sigma_{a}^{2} + {\sigma_{b}^{2}\lbrack {1 + \varphi^{2} - {2\; \varphi \; {\cos ( \frac{2\; \pi \; n}{N} )}}} \rbrack}}}}},} & (36)\end{matrix}$

so that the conditional probability for the measured trajectory,{y_(j)}, is

$\begin{matrix}{{p( { \{ y_{j} \} \middle| \varphi ,\sigma_{a}^{2},\sigma_{b}^{2}} )} = {( {2\; \pi} )^{- \frac{N}{2}}{\exp ( {{- \frac{1}{2\; \sigma_{b}^{2}}}{\sum\limits_{n = 1}^{N}y_{n}^{2}}} )} \times {\prod\limits_{n = 1}^{N}\; {\{ {\sigma_{a}^{2} + {\sigma_{b}^{2}\lbrack {1 + \varphi^{2} - {2\; \varphi \; {\cos ( \frac{2\; \pi \; n}{N} )}}} \rbrack}} \}^{- \frac{1}{2}} \times {{\exp( {\frac{1}{2\; \sigma_{b}^{2}}\frac{1}{N}{\sum\limits_{m = 1}^{N}\frac{y_{m}^{2}\sigma_{a}^{2}}{\sigma_{a}^{2} + {\sigma_{b}^{2}\lbrack {1 + \varphi^{2} - {2\; \varphi \; {\cos ( \frac{2\; \pi \; m}{N} )}}} \rbrack}}}} )}.}}}}} & (37)\end{matrix}$

This can be inverted to obtain the likelihood function for φ, σ_(a) ²and σ_(b) ²:

$\begin{matrix}{{L( {\varphi,\sigma_{a}^{2}, \sigma_{b}^{2} \middle| \{ y_{i} \} } )} = {{{- \frac{N}{2}}\ln \; 2\; \pi} - {\frac{1}{2\; \sigma_{b}^{2}}{\sum\limits_{j = n}^{N}y_{n}^{2}}} - {\frac{1}{2}{\sum\limits_{n = 1}^{N}{\ln ( {\sigma_{a}^{2} + {\sigma_{b}^{2}\lbrack {1 + \varphi^{2} - {2\; \varphi \; {\cos ( \frac{2\; \pi \; n}{N} )}}} \rbrack}} )}}} + {\frac{\sigma_{a}^{2}}{2\; \sigma_{b}^{2}}\frac{1}{N}{\sum\limits_{n = 1}^{N}{\frac{y_{n}^{2}\sigma_{a}^{2}}{\sigma_{a}^{2} + {\sigma_{b}^{2}\lbrack {1 + \varphi^{2} - {2\; \varphi \; {\cos ( \frac{2\; \pi \; n}{N} )}}} \rbrack}}.}}}}} & (38)\end{matrix}$

Best estimates ({circumflex over (φ)}, {circumflex over (σ)}_(a) ²,{circumflex over (σ)}_(b) ²) for the parameters (φ, σ_(a) ², σ_(b) ²)are solutions of the coupled equations

$\begin{matrix}{\frac{\partial L}{\partial\varphi} = {\frac{\partial L}{\partial\sigma_{a}^{2}} = {\frac{\partial L}{\partial\sigma_{b}^{2}} = 0.}}} & (39)\end{matrix}$

Eq. (39) can be solved in closed form if σ_(b) ²=0. In this case, thebest estimates for the parameters are

$\begin{matrix}{{{\hat{\varphi}}_{0} = \frac{c_{1}}{c_{2}}},{and}} & (40) \\{{{\hat{\sigma}}_{a\; 0}^{2} = {c_{0}\lbrack {1 - ( \frac{c_{1}}{c_{2}} )^{2}} \rbrack}},} & (41) \\{where} & \; \\{c_{m} = {\frac{1}{N}{\sum\limits_{j = 1}^{N}{y_{j}{y( {j + m} )}{mod}\; N}}}} & (42)\end{matrix}$

is the barrel autocorrelation of {y_(j)} at lag m. The associatedstatistical uncertainties are

$\begin{matrix}{{{\Delta \; {\hat{\varphi}}_{0}} = \sqrt{\frac{{\hat{\sigma}}_{a\; 0}^{2}}{{Nc}_{0}}}},{and}} & (43) \\{{\Delta \; {\hat{\sigma}}_{a\; 0}^{2}} = {{\hat{\sigma}}_{a\; 0}^{2}{\sqrt{\frac{2}{N}}.}}} & (44)\end{matrix}$

In the absence of measurement errors, just two descriptors, c₀ and c₁,contain all of the information that can be extracted from the timeseries regarding φ and σ_(a) ². These are examples of sufficientstatistics that completely specify the system's dynamics.

The analysis is less straightforward when σ_(b) ²≠0 because Eqs. (39) nolonger are simply separable. The system of equations can be solved atleast approximately provided the measurement error σ_(b) ² is smallerthan σ_(a) ². In this case, the best estimates for the parameters can beexpressed in terms of the error-free estimates as

$\begin{matrix}{{\hat{\varphi} \approx {{\hat{\varphi}}_{0}\{ {1 + {\frac{\sigma_{b}^{2}}{{\hat{\sigma}}_{a\; 0}^{2}}\lbrack {1 - {\hat{\sigma}}_{0}^{2} + \frac{c_{2}}{c_{0}}} \rbrack}} \}}},{and}} & (45) \\{{{\hat{\sigma}}_{a}^{2} \approx {{\hat{\sigma}}_{a\; 0}^{2} - {\frac{\sigma_{b}^{2}}{{\hat{\sigma}}_{a\; 0}^{2}}{c_{0}\lbrack {1 - {5\; {\hat{\varphi}}_{0}^{4}} + {4\; {\hat{\varphi}}_{0}^{2}\frac{c_{2}}{c_{0}}}} \rbrack}}}},} & (46)\end{matrix}$

to first order in σ_(b) ²/σ_(a) ², with statistical uncertaintiespropagated in the conventional manner. The sufficient statistics at thislevel of approximation include just one additional moment, c₂.Expansions to higher order in σ_(b) ²/σ_(a) ² require additionalcorrelations to be completed, and the exact solution requirescorrelations at all lags m. Such a complete analysis offers nocomputational benefits over power spectral analysis, for example. Itdoes, however, provide a systematic approach to estimating experimentaluncertainties. If the measurement error is small enough for Eqs. (45)and (46) to apply, the computational savings can be substantial, and theamount of data required to achieve a desired level of accuracy in thephysically relevant quantities, κ and γ, can be reduced dramatically.

The errors in locating colloidal particles' centroids can be calculatedfrom knowledge of the images' signal to noise ratio and the opticaltrain's magnification. Centroid resolutions of 10 nm or better can beroutinely attained for micrometer-scale colloidal particles in waterusing conventional bright-field imaging. In practice, however,mechanical vibrations, video jitter and other processes may increase themeasurement error by amounts that can be difficult to independentlyquantify. Quite often, the overall measurement error is most easilyassessed by increasing the laser power to the optical traps to minimizethe particles' thermally driven motions. In this case, y_(j)≈b_(j), andthe measurement error's variance σ_(b) ² can be estimated directly.

$\begin{matrix}{{\frac{\kappa}{k_{B}T} = \frac{1 - {\hat{\varphi}}^{2}}{{\hat{\sigma}}_{a}^{2}}},{and}} & (47) \\{{\frac{\gamma}{k_{B}T\; \Delta \; t} = {- \frac{1 - {\hat{\varphi}}^{2}}{\sigma_{a}^{2}\ln \; \hat{\varphi}}}},} & (48)\end{matrix}$

with error estimates, Δκ and Δγ, given by

$\begin{matrix}{( \frac{\Delta \; \kappa}{\kappa} )^{2} = {( \frac{\Delta \; {\hat{\sigma}}_{a}^{2}}{{\hat{\sigma}}_{a}^{2}} ) + {( \frac{2\; {\hat{\varphi}}^{2}}{1 - {\hat{\varphi}}^{2}} )^{2}( \frac{\Delta \; \hat{\varphi}}{\hat{\varphi}} )^{2}}}} & (49) \\{( \frac{\Delta \; \gamma}{\gamma} )^{2} = {( \frac{\Delta \; {\hat{\sigma}}_{a}^{2}}{{\hat{\sigma}}_{a}^{2}} )^{2} + {( {\frac{2\; {\hat{\varphi}}^{2}}{1 - {\hat{\varphi}}^{2}} - \frac{1}{\ln \; \hat{\varphi}}} )^{2}{( \frac{\Delta \; \hat{\varphi}}{\hat{\varphi}} )^{2}.}}}} & (50)\end{matrix}$

If the measurement interval Δt is much longer than the viscousrelaxation time τ=γ/κ, then φ vanishes and the error in the dragcoefficient diverges. Conversely, if Δt is much smaller than τ, then φapproaches unity and both errors diverge. Consequently, this approach totrap characterization does not benefit from excessively fast sampling.Rather, it relies on accurate particle tracking to minimize Δ{circumflexover (φ)} and Δ{circumflex over (σ)}_(a) ². For trap-particlecombinations with viscous relaxation tunes of several milliseconds orlonger and typical particle excursions of at least 10 nm, digital videomicroscopy provides both the temporal and spatial resolution needed tocompletely characterize optical traps. This approach also lends itselfto simultaneous characterization of multiple traps, which is notpossible with conventional methods.

In the event that measurement errors can be ignored (σ_(b) ²<<σ_(a) ²),the physically relevant quantities can be obtained as:

$\begin{matrix}{\frac{\kappa_{0}}{k_{B}T} = {\frac{1}{c_{0}}\lbrack {1 \pm \sqrt{\frac{2}{N}( {1 + \frac{2\; c_{1}^{2}}{c_{0}^{2} - c_{1}^{2}}} )}} \rbrack}} & (51) \\{{{\frac{\gamma \; 0}{k_{B}T\; \Delta \; t} = {\frac{1}{c_{0}{\ln ( \frac{c_{0}}{c_{1}} )}}( {1 \pm \frac{\Delta \; \gamma \; 0}{\gamma \; 0}} )}},{where}}{{N( \frac{{\Delta \; \gamma_{0}}\;}{\gamma_{0}} )}^{2} = {2 + {{\frac{1}{c_{0}^{2} - c_{1}^{2}}\lbrack \frac{c_{1}^{2} - {2\; c_{1}c_{0}{\ln ( \frac{c_{0}}{c_{1}} )}} - c_{0}^{2}}{c_{0}{\ln ( \frac{c_{0}}{c_{1}} )}} \rbrack}^{2}.}}}} & (52)\end{matrix}$

These results are not reliable if c₁≦σ_(b) ², which when the samplinginterval Δt is much longer or shorter than the viscous relaxation time,τ. Accurate estimates for κ and γ still may be obtained in this case byapplying Eqs. (45) and (46) or their generalizations.

Optimal statistical analysis offers insights not only into the traps'properties, but also into the properties of the trapped particles andthe surrounding medium. For example, if a spherical probe particle isimmersed in a medium of viscosity η far from any bounding surfaces, itshydrodynamic radius a can be assessed from the measured drag coefficientusing the Stokes result γ=6^(π)ηa. The viscous drag coefficients alsoprovide insights into the particles' coupling to each other and to theirenvironment. The independently assessed values of the traps' stiffnessthen can serve as a self-calibration in microrheological measurementsand in measurements of colloidal many-body hydrodynamic coupling. Incases where the traps themselves must be accurately calibrated,knowledge of the probe particles' differing properties gauged frommeasurements of γ can be used to distinguish variations in the traps'intrinsic properties from variations due to differences among the probeparticles. The apparent width and depth of the potential energy well aparticle experiences when it encounters an optical trap depends on itssize in a manner that can be inverted at least approximately.

These measurements, moreover, can be performed rapidly enough, even atconventional video sampling rates, to permit real-time adaptiveoptimization of the traps' properties. Each trap's stiffness is roughlyproportional to its brightness. Therefore, if the m-th trap in an arrayis intended to receive a fraction |αm|₂ of the projected light, then itsstiffness should satisfy

$\begin{matrix}{\frac{\kappa_{m}}{\sum\limits_{i = 1}^{N}\kappa_{i}} = {\alpha_{m}}^{2}} & (53)\end{matrix}$

Any departure from this due to fixed instrumental deficiencies can becorrected by modifying the design amplitudes,

$\begin{matrix} \alpha_{m}arrow{\alpha_{m}\sqrt{\frac{\sum\limits_{i = 1}^{N}\kappa_{i}}{\kappa_{m}},}}  & (54)\end{matrix}$

and recalculating the CGH.

As a practical demonstration of the utility of the present invention, achallenging pattern of optical traps is calculated, projected, andcharacterized in a quasiperiodic array. The traps are formed with a100×NA 1.4 S-Plan Apo oil immersion objective lens mounted in a NikonTE-2000U inverted optical microscope. The traps are powered by aCoherent Verdi frequency-doubled diode-pumped solid state laseroperating at a wavelength of 532 nm. Computer-generated phase hologramsare imprinted on the beam with a Hamamatsu X8267-16 parallel-alignednematic liquid crystal spatial light modulator (SLM). This SLM canimpose phase shifts up to 2^(π) radians at each pixel in a 760×760array. The face of the SLM is imaged onto the objective's 5 mm diameterinput pupil using relay optics designed to minimize aberrations. Thebeam is directed into the objective with a dichroic beamsplitter (ChromaTechnologies), which allows images to pass through to a low-noisecharge-coupled device (CCD) camera (NEC TI-324AII). The video stream isrecorded as uncompressed digital video with a Pioneer digital videorecorder (DVR) for processing.

FIG. 4( a) shows the intended planar arrangement of 119 holographicoptical traps. Even after adaptive-additive refinement, the hologramresulting from simple superposition with random phase fares poorly forthis aperiodic pattern. FIG. 4( b) shows the intensity of lightreflected by a front-surface mirror placed in the sample plane. Thisimage reveals several undesirable defects including extraneous ghosttraps, an exceptionally bright central spot, and large variability inintensity. Imaging photometry on this and equivalent images producedwith different random relative phases for the beams yields a typicalroot-mean-square variation of more than 50 percent in the projectedtraps' brightness. The image in FIG. 4( c) was produced using themodified optical train and the direct search algorithm describedearlier. This image suffers from none of these defects exemplified inFIG. 4( b). Both the ghost traps and the central spot are suppressed,and the apparent relative brightness variations are smaller than 5percent, an improvement by a factor of ten.

The real test of these optical tweezers, however, is their performanceat trapping particles. FIG. 4( d) shows 119 colloidal silica spheres,2a=1.6±0.2 μm pin in diameter, dispersed in water at T=40° C. Theviscosity is roughly η=1 cP. The dispersion was sealed into a slit poreformed by sealing the edges of a glass #1.5 cover slip to the surface ofa glass microscope slide. The array of traps was focused roughly 10 μmabove the inner surface of the coverslip in a layer roughly 40 μm thick.The traps were separated by 7 μm, so that hydrodynamic coupling amongthe spheres should modify their individual drag coefficients by no morethan ten percent, which is comparable to the effect of the nearby wall.Imaging spheres in smaller trapping patterns at a projected power of 30mW per trap suggests that the overall measurement error for theparticles' centroids is σ_(b) ²=5 nm².

Reducing the laser power to 2 mW per trap frees the particles to explorethe traps' potential energy wells. The particles were tracked both alongthe line of traps (the {circumflex over (x)} direction) andperpendicular to it (the ŷ direction), and analyzed both coordinatesseparately, using the methods discussed herein. It was shown that thetraps' strengths do indeed vary by more than the typical measurementerror, but that the variation is less than 5 percent. If the variationswere larger, information from this measurement could be used to adjustthe amplitudes am in Eq. (5) to correct for fixed variations in theoptical train's performance.

B. Flux Reversal in Symmetric Optical Thermal Ratchets

The ability to rectify Brownian forces with spatially extendedtime-varying light fields creates new opportunities for leveraging thestatistical properties of thermal ratchets and to exploit them by theirinteresting and useful properties for practical applications. In theseembodiments a one-dimensional thermal ratchet implemented with theholographic optical trapping technique applied to fluid-borne colloidalspheres. The complementary roles of global spatiotemporal symmetry andlocal dynamics are presented in relation to establishing the directionof ratchet-induced motion and also present applications inhigher-dimensional systems.

Thermal ratchets employ time-varying potential energy landscapes tobreak the spatiotemporal symmetry of thermally equilibrated systems. Theresulting departure from equilibrium takes the form of a directed fluxof energy or materials, which can be harnessed for natural and practicalapplications. Unlike conventional macroscopic machines whose efficiencyis reduced by random fluctuations, thermal ratchets actually can utilizenoise to operate. They achieve their peak efficiency when their spatialand temporal evolution is appropriately matched to the scale offluctuations in the heat bath.

Most thermal ratchet models involve locally asymmetric space-fillingpotential energy landscapes, and almost all are designed to operate inone dimension. Most practical implementations have exploitedmicrofabricated structures such as interdigitated electrode arrays,quantum dot arrays, periodic surface textures, or microfabricated poresfor hydrodynamic drift ratchets. Previous optical implementations haveused a rapidly scanned optical tweezer to create an asymmetricone-dimensional potential energy landscape in a time-averaged sense, ora time-varying dual-well potential with two conventional optical traps.

This embodiment includes a broad class of optical thermal ratchets thatexploit the holographic optical tweezer technique to create large-scaledynamic potential energy landscapes. This approach permits detailedstudies of the interplay of global spatiotemporal symmetry and localdynamics in establishing both the magnitude and direction ofratchet-induced fluxes. It also provides for numerous practicalapplications.

Holographic optical tweezers use computer-generated holograms to projectlarge arrays of single-beam optical traps. One implementation, shownschematically in FIG. 5( a), uses a liquid crystal spatial lightmodulator 100 (SLM) (Hamamatsu X7550 PAL-SLM) to imprint phase-onlyholograms on the wavefronts of a laser beam 102 from a frequency-doubleddiode-pumped solid state laser 104 operating at 532 nm (Coherent Verdi).This SLM 100 can vary the local phase, φ(r), between 0 and 2π radians ateach position r in a 480×480 grid spanning the beam's wavefront. Amodulated beam 106 is relayed to the input pupil of a 100×NA 1.4 SPlanApo oil immersion objective lens 108 mounted in an inverted opticalmicroscope 110 (Zeiss S-100TV). The objective lens 108 focuses the lightinto a pattern of optical traps that can be updated in real time bytransmitting a new phase pattern to the SLM.

FIG. 5( b) shows the focused light, l({right arrow over (r)}), from atypical pattern of holographic optical traps, which is imaged by placinga front-surface mirror on the sample stage and collecting the reflectedlight with the objective lens 108. Each focused spot of light in this20×5 array constitutes a discrete optical tweezer, which acts as aspatially symmetric three-dimensional potential energy well for amicrometer-scale object. FIG. 5( b) shows an aqueous dispersion of 1.53μm diameter colloidal silica spheres (Bangs Laboratories, lot number5328) interacting with this pattern of traps at a projected laser powerof 2.5 mW/trap.

Each potential well may be described as a rotationally symmetricGaussian potential well. Arranging the traps in closely spaced manifoldsseparated by a distance L creates a pseudo-one-dimensional potentialenergy landscape, V(x), which can be modeled as

$\begin{matrix}{{V(x)} = {{- V_{0}}{\sum\limits_{n = {- N}}^{N}{{\exp( {- \frac{( {x - {nL}} )^{2}}{2\; \sigma^{2}}} )}.}}}} & (55)\end{matrix}$

The well depth, V₀, approaches the thermal energy scale, β⁻¹, when eachoptical tweezer is powered with somewhat less than 1 mW of light. Theholographically projected traps' strengths are uniform to within tenpercent. Their widths, σ are comparable to the spheres' radii. With thetraps powered by 3 mW, diffusing particles are rapidly localized by thefirst optical tweezer they encounter, as can be seen from the centerphotograph in FIG. 5( c).

The potential energy landscape created by a holographic optical tweezerarray differs from most ratchet potentials in two principal respects. Inthe first place, the empty spaces between manifolds comprise largeforce-free regions. This contrasts with most models, which employspace-filling landscapes. The landscape can induce motion only if randomthermal fluctuations enable particles to diffuse across force-freeregions. Secondly, the landscape is spatially symmetric, both globallyand locally. Breaking spatiotemporal symmetry to induce a flux rests,therefore, with the landscape's time evolution. Details of the protocolcan determine the nature of the induced motion.

The most straightforward protocols for holographic optical thermalratchets involve cyclically translating the landscape by discretefractions of the lattice constant L, with the n-th state in each cyclehaving duration T_(n). The motion of a Brownian particle in such asystem can be described with the one-dimensional Langevin equation

γ{dot over (x)}(t)=−V′(x(t)−ƒ(t))+ξ(t),  (56)

where y is the particle's viscous drag coefficient, the prime denotes aderivative with respect to the argument, and ξ(t) is a stochastic forcerepresenting thermal noise. This white-noise forcing satisfies

ξ(t)

=0 and

ξ(t)ξ(s)

=2(y|β)δ(t−s).

The potential energy landscape in our system is spatially periodic:

V(x+L)=V(x).  (57)

The discrete displacements in an N-state cycle, furthermore, also aredescribed by a periodic function ƒ(t), with period

$T = {\sum\limits_{n - 1}^{N}{T_{n}.}}$

That a periodically driven, symmetric and spatially periodic potentialcan rectify Brownian motion to generate a directed flux might not beimmediately obvious. Directed motion in time-evolving landscapes is allbut inevitable, with flux-free operation being guaranteed only if V(x)and ƒ(t) satisfy specific conditions of spatiotemporal symmetry,

V(x)=V(−x), and {dot over (ƒ)}(t)=−{dot over (ƒ)}(t+T/2),  (58)

and spatiotemporal supersymmetry,

V(x)=−V(x+L/2), and {dot over (ƒ)}(t+Δt)=−{dot over (ƒ)}(−t),  (59)

for at least one value of Δt. The dot in Eqs. (58) and (59) denotes atime derivative. Two distinct classes of one-dimensional optical thermalratchets that exploit these symmetries in different ways are presentedherein. The first results in directed diffusion except for a particularoperating point, at which Eq. (58) is satisfied. The second has a pointof flux-free operation even though Eqs. (58) and (59) are alwaysviolated. In both cases, the vanishing point signals a reversal of thedirection of the induced flux.

The simplest optical ratchet protocol involves a two-state cycle,

$\begin{matrix}{{f(t)} = \{ \begin{matrix}{0,} & {0 \leq ( {t\; {mod}\; T} ) < T_{1}} \\\frac{L}{3} & {T_{1} \leq ( {t\; {mod}\; T} ) < T}\end{matrix} } & (60)\end{matrix}$

This protocol explicitly satisfies the symmetry condition in Eq. (58)when the two states are of equal duration, T₁=T₂=T/2. This particularoperating point therefore should create a flux-free nonequilibriumsteady-state, with particles being juggled back and forth betweenneighboring manifolds of traps. Breaking spatiotemporal symmetry bysetting T₁≠T₂ does not guarantee a flux, but at least creates thepossibility.

FIG. 6 presents flux induced by a two-state holographic optical ratchet.Discrete points show measured mean drift speed as a function of T₂ forT₁=3 sec. The solid curve is a fit to the data for βV₀=2.75 and σ=0.65μm. Other curves show how the induced drift depends on T/τ, with optimalflux obtained for T/τ=0.193.

The data in FIG. 6 demonstrate that this possibility is borne out inpractice. The discrete points in FIG. 6 show the measured average driftvelocity, v, for an ensemble of colloidal silica spheres 1.53 μm indiameter dispersed in a 40 μm thick layer of water between a coverslipand a microscope slide. The spheres are roughly twice as dense as waterand rapidly sediment into a free-floating layer above the coverslip. Theholographic optical tweezer array was projected into the layer'smidplane to minimize out-of-plane fluctuations, with an estimated powerof 1 mW/trap. Roughly 30 spheres were in the trapping domain at anytime, so that reasonable statistics could be amassed in 10 minutesdespite the very large fluctuations inherent in thermal ratchetoperation. This number is small enough, moreover, to minimize the rateof collisions among the particles.

Given the spheres' measured diffusion coefficient of D=0.33 μm²/sec.,the time required to diffuse the inter-manifold separation of L=5.2 μmis τ=L²/(2D)=39 sec. This establishes a natural velocity scale, L/τ, inwhich v is presented. These data were acquired with T₁=3 sec. and T₂varying from 0.8 sec to 14.7 sec.

As anticipated, the ratchet-induced flux vanishes at the point ofspatiotemporal symmetry, T₂=T₁, and is non-zero otherwise. The vanishingpoint signals a reversal in the direction of the drift velocity, withparticles being more likely to advance from the wells in thelonger-lived state toward the nearest manifold in the shorter-livedstate. This trend can be understood as resulting from the short-durationstate's biasing the diffusion of particles away from their localizeddistribution in the long-lived state.

To make this qualitative argument more concrete, it is possible tocalculate the steady-state velocity for particles in this system byconsidering the evolution of the probability density ρ(x,t) for findinga particle within dx of position x at time t. The Fokker-Planck equationassociated with Eq. (2) is:

$\begin{matrix}{{\frac{\partial{\rho ( {x,t} )}}{\partial t} = {D\lbrack {{\frac{\partial^{2}}{\partial x^{2}}{\rho ( {x,t} )}} + {\beta \frac{\partial}{\partial x}\{ {{\rho ( {x,t} )}{V^{\prime}( {x - {f(t)}} )}} \}}} \rbrack}},} & (61)\end{matrix}$

where the prime denotes a derivative with respect to the argument.Equation (61) is formally solved by the master equation

ρ(x,t+T)=∫P(x,T|x ₀,0)ρ(x ₀ ,t)dx ₀  (62)

for the evolution of the probability density, with the propagator

$\begin{matrix}{{{P( {x, t \middle| x_{0} ,0} )} = {{\exp ( {\int_{\;}^{t}{{L( {x,t^{\prime}} )}\ {t^{\prime}}}} )}{\delta ( {x - x_{0}} )}}},} & (63)\end{matrix}$

describing the transfer of particles from x₀ to x under the Liouvilleoperator

$\begin{matrix}{{L( {x,t} )} = {{D( {\frac{\partial^{2}}{\partial x^{2}} + {\beta \frac{\partial}{\partial x}{V^{\prime}( {x - {f(t)}} )}}} )}.}} & (64)\end{matrix}$

From Eq. (62), it follows that the steady-state particle distributionρ(x) is an eigenstate of the propagator,

ρ(x)=∫P(x,T|x ₀,0)ρ(x ₀)dx ₀,  (65)

associated with one complete cycle. The associated steady-state flux is

$\begin{matrix}{v = {\int{\frac{x - x_{0}}{T}{\rho ( x_{0} )}{P( {x, T \middle| x_{0} ,0} )}{x}{{x_{0}}.}}}} & (66)\end{matrix}$

FIG. 7( a) presents Stochastic resonance in the two-state opticalthermal ratchet for σ/L=0.125 with dependence on cycle period T in unitsof the diffusive time scale τ for βV₀=2.5 at the optimal duty cycleT₂/T₁=0.3. FIG. 7( b) presents dependence on well depth for the optimalcycle rate T₂/T₁=0.193 and duty cycle.

The solid curve in FIG. 6 is a fit of Eq. (66) to the measured particlefluxes for βV₀=2.5 and σ=0.65 μm. The additional curves in FIG. 6 showhow v varies with T₂/T₁ for various values of T/τ for these controlparameters. The induced flux, v, plotted in FIG. 7( a), falls off as 1/Tin the limit of large T because the particles spend increasingly much oftheir time localized in traps. It also vanishes in the opposite limitbecause the diffusing particles cannot keep up with the landscape'sevolution. The optimal cycle period at T/τ≈0.2 constitutes an example ofstochastic resonance. Although a particle's diffusivity controls thespeed with which it traverses the ratchet, its direction is uniquelydetermined by T₂/T₁.

No flux results if the traps are too weak. Increasing the potentialwells' depths increases the maximum attainable flux, but only up to apoint. If the traps are too strong, particles also become localized inthe short-lived state, and the ratchet approaches a deterministicflux-free limit in which particles simply hop back and forth betweenneighboring manifolds. This behavior is shown in FIG. 7( b).

Different objects exposed to the same time-evolving optical intensitypattern experience different values of V₀ and σ, and also can havediffering diffusive time scales, τ. Such differences establish adispersion of mean velocities for mixtures of particles moving throughthe landscape that can be used to sorting the particles. Despite thismethod's symmetry and technical simplicity, however, the two-stateprotocol is not the most effective platform for such practicalapplications. A slightly more elaborate protocol yields a thermalratchet whose deterministic limit transports material rapidly and whosestochastic limit yields flux reversal at a point not predicted by thesymmetry selection rules in Eqs. (58) and (59).

In another embodiment, the next step up in complexity and functionalrichness involves the addition of a third state to the ratchet cycle:

$\begin{matrix}{{f(t)} = {\begin{Bmatrix}{0,{0 \leq ( {t\; {mod}\; T} ) < \frac{T}{3}}} \\{\frac{L}{3},{\frac{T}{3} \leq ( {t\; {mod}\; T} ) < \frac{2\; T}{3}}} \\{{- \frac{L}{3}},{\frac{2\; T}{3} \leq ( {t\; {mod}\; T} ) < T}}\end{Bmatrix}.}} & (67)\end{matrix}$

This three-state cycle consists of cyclic displacements of the landscapeby one third of a lattice constant. Unlike the two-state symmetricthermal ratchet, it has a deterministic limit, an explanation of whichhelps to elucidate its operation in the stochastic limit.

If the width, σ, of the individual wells is comparable to the separationL/3 between manifolds in consecutive states, then a particle localizedat the bottom of a well in one state is released near the edge of a wellin the next. Provided V₀ is large enough, the particle falls to thebottom of the new well during the T/3 duration of the new state. Thisprocess continues through the sequence of states, and the particle istransferred deterministically forward from manifold to manifold. Thisdeterministic process is known as optical peristalsis, and is useful forreorganizing fluid-borne objects over large areas with simple sequencesof generic holographic trapping patterns.

Assuming the individual traps are strong enough, optical peristalsistransfers objects forward at speed v=L/T. If, on the other hand, βV₀<1,particles can be thermally excited out of the forward-going wave oftraps and so will travel forward more slowly. This is an example of adeterministic machine's efficiency being degraded by thermalfluctuations. This contrasts with the two-state thermal ratchet, whichhas no effect in the deterministic limit and instead relies on thermalfluctuations to induce motion.

The three-state protocol enters its stochastic regime when theinter-state displacement of manifolds, L/3, exceeds the individualtraps' width, σ. Under these conditions, a particle that is trapped inone state is released into the force-free region between traps once thestate changes. If the particle diffuses rapidly enough, it mightnevertheless fall into the nearest potential well centered a distanceL/3 away in the forward-going direction within time T/3. The fraction ofparticles achieving this will be transferred forward in each step of thecycle. This stochastic process resembles optical peristalsis, albeitwith reduced efficiency. There is a substantial difference, however.

An object that does not diffuse rapidly enough to reach the nearestforward-going trap in time T/3 might still reach the trap centered at−L/3 in the third state by time 2T/3. Such a slow-moving object would betransferred backward by the ratchet at velocity v=−L/(2T). Unlike thetwo-state ratchet, whose directionality is established unambiguously bythe sequence of states, the three-state ratchet's direction appears todepend also on the transported objects' mobility.

FIG. 8( a) presents flux reversal in a symmetric three-state opticalthermal ratchet and depicts this as a function of cycle period for fixedinter-manifold separation, L. FIG. 8( b) shows this as a function ofinter-manifold separation L for fixed cycle period T.

These observations above are borne out by the experimental observationsin FIGS. 8( a) and 8(b). The discrete points in FIG. 8( a) show themeasured flux of 1.53 μm diameter silica spheres as a function of thecycle period T with the inter-manifold separation fixed at L=6.7 μm.Flux reversal at T/τ=0.1 does not result from special symmetryconsiderations because the spatiotemporal evolution described by Eqs.(55) and (67) violates the conditions in Eqs. (58) and (59) for allvalues of T. Rather, this reflects a dynamical transition in whichrapidly diffusing particles are driven in the forward while slowlydiffusing particles drift backward. The origin of this transition inthermal ratchet behavior is confirmed by the observation of a comparabletransition induced by varying the inter-manifold separation L for fixedcycle period T, as plotted in FIG. 8( b).

FIG. 9 presents calculated ratchet-induced drift velocity as a functionof cycle period T for representative values of the inter-manifoldseparation L ranging from the deterministic limit, L=6.5σ to thestochastic limit L=13σ.

The solid curves in FIGS. 8( a) and 8 (b) are fits to Eq. (66) using Eq.(67) to calculate the propagator. The fit values, βV₀=8.5±0.08 andσ=0.53±0.01 μm are consistent with values obtained for the two-stateratchet, given a higher laser power of 2.5 mW/trap. The crossover fromdeterministic optical peristalsis with uniformly forward-moving flux atsmall L to stochastic operation with flux reversal at larger separationsis captured in the calculated drift velocities plotted in FIG. 9.

Whereas flux reversal in the two-state ratchet is mandated by theprotocol, flux reversal in the three-state ratchet depends on propertiesof diffusing objects through the detailed structure of the probabilitydistribution ρ(x) under different operating conditions. The three-stateoptical thermal ratchet therefore provides the basis for sortingapplications in which different fractions of a mixed sample aretransported in opposite directions by a single time-evolving opticallandscape. This builds upon previously reported ratchet-basedfractionation techniques which rely on unidirectional motion.

C. Radial Ratchet

The flexibility of holographic optical thermal ratchet implementationsand the success of our initial studies of one-dimensional variants bothinvite consideration of thermal ratchet operation in higher dimensions.This is an area that has not received much attention, perhaps because ofthe comparative difficulty of implementing multidimensional ratchetswith other techniques. As an initial step in this direction, it ispossible to introduce a ratchet protocol in which manifolds of traps areorganized into evenly spaced concentric rings whose radii advancethrough a three-state cycle analogous to that in Eq. (67). Theprobability distribution p(r,t) for a Brownian particle to be foundwithin dr of r at time t under external force F(r,t)=−∇V(r,t) satisfies

$\begin{matrix}{\frac{\partial{p( {r,t} )}}{\partial t} = {{D\lbrack {{\nabla^{2}{p( {r,t} )}} - {\beta \; {\nabla{\cdot \{ {{p( {r,t} )}{F( {r,t} )}} \}}}}} \rbrack}.}} & (68)\end{matrix}$

If the force depends only on the radial coordinate asF(r,t)−∂V(r,t){circumflex over (r)}, Eq. (68) reduces to

$\begin{matrix}{\frac{\partial{p( {r,t} )}}{\partial t} = {{D\lbrack {{\frac{1}{r}\frac{\partial}{\partial r}\{ {r\frac{\partial}{\partial r}{p( {r,t} )}} \}} + {\frac{\beta}{r}\frac{\partial}{\partial r}\{ {{{rV}^{\prime}( {r,t} )}{p( {r,t} )}} \}}} \rbrack}.}} & (69)\end{matrix}$

The probability p(r,t) for a particle to be found between r and r+dr attime t is given by p(r,t)=2πrp(r,t). Therefore, the Fokker-Planckequation can be rewritten in terms of p(r,t) as

$\begin{matrix}{\frac{\partial{\rho ( {r,t} )}}{\partial t} = {{D\lbrack {{\frac{\partial^{2}}{\partial r^{2}}{\rho ( {r,t} )}} + {\beta \frac{\partial}{\partial r}\{ {( {{V^{\prime}( {r,t} )} - \frac{1}{\beta \; r}} ){\rho ( {r,t} )}} \}}} \rbrack}.}} & (70)\end{matrix}$

This, in turn, can be reduced to the form of Eq. (61) by introducing theeffective one-dimensional potential V_(eff)(r,t)≡V(r−ƒ(t))−β⁻¹1 nr. Therest of the analysis follows by analogy to the linear three-stateratchet.

FIGS. 10( a)-10(d) present fractionation in a radial optical thermalratchet. FIG. 10( a) presents the pattern of concentric circularmanifolds with L=4.7 μm. FIG. 10( b) presents a mixture of large andsmall particles interacting with a fixed trapping pattern. FIG. 10( c)presents small particles collected and large excluded at L=4.9 μm andT=4.5 sec. FIG. 10( d) presents large particles concentrated at L=5.3 μmand T=4.5 sec. The scale bar indicates 10 μm.

Like the linear variant, the three-state radial ratchet has adeterministic operating regime in which objects are clocked inward oroutward depending on the sequence of states. The additional geometricterm in V_(eff)(r) and the constraint that r>0 substantially affect theradial ratchet's operation in the stochastic regime by inducing aposition-dependent outward drift. In particular, a particle being drawninward by the ratchet effect must come to a rest at a radius where theratchet-induced flux is balanced by the geometric drift. Outward-drivenparticles, by contrast, are excluded by the radial ratchet. Combiningthis effect with the three-state ratchet's natural propensity formobility-dependent flux reversal suggests that radial ratchet protocolscan be designed to sort mixtures in the field of view, expelling theunwanted fraction and concentrating the target fraction. This behavioris successfully demonstrated in FIG. 10( c), in which 1 μm diametersilica spheres (Bangs Laboratories, lot number 21024) have beencollected within an outward-driving radial ratchet at L=4.9 μm at T−4.5sec. while larger 1.53 μm diameter silica spheres are expelled, and inFIG. 10( d), in which the opposite is achieved with an inward-drivingratchet at L=5.3 μm and the same period, T=4.5 sec. A larger and morerefined version might sort different fractions into concentric ringswithin the ratchet domain. This capability might find applications inisolating and identifying individual bacterial species within biofilms,for example.

This embodiment of the invention provides for one-dimensional thermalratchet models implemented with holographic optical tweezer arrays. Theuse of discrete optical tweezers to create extensive potential energylandscapes characterized by large numbers of locally symmetric potentialenergy wells provides a practical method for thermal ratchet behavior tobe induced in large numbers of diffusing objects in comparatively largevolumes. The particular applications described herein all can be reducedto one-dimensional descriptions, and are conveniently analyzed with theconventional Fokker-Planck formalism. In each case, the ratchet-induceddrift is marked by an operating point at which the flux reverses. Insymmetric two-state traveling ratchets, flux reversal occurs at a pointpredicted by Reimann's symmetry selection rules. The three-statevariants, on the other hand, undergo flux reversal as a consequence of acompetition between the landscapes' temporal evolution and the Brownianparticles' diffusion. The latter mechanism, in particular, suggestsopportunities for practical sorting applications.

The protocols described herein can be generalized in several ways. Thedisplacements between states, for example, could be selected to optimizetransport speed or to tune the sharpness of the flux reversal transitionfor sorting applications. Similarly, the states in our three-stateprotocol need not have equal durations. They also might be tuned tooptimize sorting, and perhaps to select a particular fraction from amixture. The limiting generalization is a pseudo-continuous travelingratchet with specified temporal evolution, ƒ(t). The present embodimentuses manifolds of traps which may all be of the same geometry andintensity. The present invention may also use manifolds of traps wherethe geometry and intensity are not all the same. These characteristicsalso can be specified, with further elaborations yielding additionalcontrol over the ratchet-induced transport. The protocols described hereare useful for dealing with the statistical mechanics of symmetrictraveling ratchets and may be used in practical applications.

Just as externally driven colloidal transport through statictwo-dimensional arrays of optical traps gives rise to a hierarchy ofkinetically locked-in states, ratchet-induced motion throughtwo-dimensional and three-dimensional holographic optical tweezer arraysis likely to be complex and interesting. Various other proposedhigher-dimensional ratchet models have been experimentally implemented.None of these has explored the possibilities of scaling ratchetsresembling the radial ratchet introduced here but with irreducible two-or three-dimensional structure.

D. Flux Reversal in Three State Thermal Ratchets

A cycle of three holographic optical trapping patterns can implement athermal ratchet for diffusing colloidal spheres, and the ratchet-driventransport displays flux reversal as a function of the cycle frequencyand the inter-trap separation. Unlike previously described ratchetmodels, the present invention involves three equivalent states, each ofwhich is locally and globally spatially symmetric, with spatiotemporalsymmetry being broken by the sequence of states.

Brownian motion cannot create a steady flux in a system at equilibrium.Nor can local asymmetries in a static potential energy landscape rectifyBrownian motion to induce a drift. A landscape that varies in time,however, can eke a flux out of random fluctuations by breakingspatiotemporal symmetry. Such flux-inducing time-dependent potentialsare known as thermal ratchets, and their ability to bias diffusion byrectifying thermal fluctuations has been proposed as a possiblemechanism for transport by molecular motors and is being activelyexploited for macromolecular sorting.

Most thermal ratchet models are based on spatially asymmetricpotentials. Their time variation involves displacing or tilting themrelative to the laboratory frame, modulating their amplitude, changingtheir periodicity, or some combination, usually in a two-state cycle.Thermal ratcheting in a spatially symmetric double-well potential hasbeen demonstrated for a colloidal sphere in a pair of intensitymodulated optical tweezers. More recently, directed transport has beeninduced in an atomic cloud by a spatially symmetric rocking ratchetcreated with an optical lattice.

The space-filling potential energy landscapes required for most suchmodels pose technical challenges. Furthermore, their relationship to theoperation of natural thermal ratchets has not been resolved. In thisembodiment of the invention, a spatially symmetric thermal ratchet isshown implemented with holographic optical traps. The potential energylandscape in this system consists of a large number of discrete opticaltweezers, each of which acts as a symmetric potential energy well fornanometer- to micrometer-scale objects such as colloidal spheres. Thesewells are arranged so that colloidal spheres can diffuse freely in theinterstitial spaces but are localized rapidly once they encounter atrap. A three-state thermal ratchet then requires only displaced copiesof a single two-dimensional trapping pattern. Despite its simplicity,this ratchet model displays flux reversal in which the direction ofmotion is controlled by a balance between the rate at which particlesdiffuse across the landscape and the ratchet's cycling rate.

FIGS. 11( a)-11(d) present a spatially-symmetric three-state ratchetpotential comprised of discrete potential wells. Flux reversal has beendirectly observed in comparatively few systems. Flux reversal arises asa consequence of stochastic resonance for a colloidal sphere hoppingbetween the symmetric double-well potential of a dual optical trap.Previous larger-scale demonstrations have focused on ratcheting ofmagnetic flux quanta through type-II superconductors in both the quantummechanical and classical regimes, or else have exploited the crossoverfrom quantum mechanical to classical transport in a quantum dot array.Unlike the present implementation, these exploit spatially asymmetricpotentials and take the form of rocking ratchets. A similarcrossover-mediated reversal occurs for atomic clouds in symmetricoptical lattices. A hydrodynamic ratchet driven by oscillatory flowsthrough asymmetric pores also shows flux reversal. In this case,however, the force field is provided by the divergence-free flow of anincompressible fluid rather than a potential energy landscape, and so isan instance of a so-called drift ratchet. Other well knownimplementations of classical force-free thermal ratchets also were basedon asymmetric potentials, but did not exhibit flux reversal.

FIGS. 11( a)-11(d) show the principle upon which the three-state opticalthermal ratchet operates. The process starts out with a pattern ofdiscrete optical traps, each of which can localize an object. Thepattern in the initial state is schematically represented as threediscrete potential energy wells, each of width σ and depth V₀, separatedby distance L. A practical trapping pattern can include a great manyoptical traps organized into manifolds. The first pattern of FIG. 11( a)is extinguished after time T and replaced immediately with the second(FIG. 11( b)), which is displaced from the first by L/3. This isrepeated in the third state with an additional step of L/3 (FIG. 11(c)), and again when the cycle is completed by returning to the firststate (FIG. 11( d)).

If the traps in a given state overlap those in the state before, atrapped particle is transported deterministically forward. Runningthrough this cycle repeatedly transfers the object in a directiondetermined unambiguously by the sequence of states, and is known asoptical peristalsis. The direction of motion can be reversed only byreversing the sequence.

The optical thermal ratchet differs from this in that the inter-trapseparation L is substantially larger than σ. Consequently, particlestrapped in the first pattern are released into a force-free region andcan diffuse freely when that pattern is replaced by the second. Thoseparticles that diffuse far enough to reach the nearest traps in thesecond pattern rapidly become localized. A comparable proportion of thislocalized fraction then can be transferred forward again once the thirdpattern is projected, and again when the cycle returns to the firststate.

Unlike optical peristalsis, in which all particles are promoted in eachcycle, the stochastic ratchet transfers only a fraction. This, however,leads to a new opportunity. Particles that miss the forward-going wavemight still reach a trap on the opposite side of their starting pointwhile the third pattern is illuminated. These particles would betransferred backward by L/3 after time 2T.

For particles of diffusivity D, the time required to diffuse theinter-trap separation is τ=L²/(2D). Assuming that particles begin eachcycle well localized at a trap, and that the traps are well separatedcompared to their widths, then the probability for ratcheting forward byL/3 during the interval T is roughly P_(F)≈exp(−(L/3)²/(2DT)), while theprobability of ratcheting backwards in time 2T is roughlyP_(R)≈exp(−(L/3)²/(4DT)). The associated fluxes of particles then arev_(F)=P_(F)L/(3T) and v_(R)=−P_(R)L/(6T), with the dominant termdetermining the overall direction of motion. The direction of inducedmotion may be expected to reverse when P_(F)≈exp(−(L/3)²/(2DT)), or forexample when T/τ<(18 ln 2)⁻¹≈0.08.

More formally, this can be modeled as an array of optical traps in then-th pattern as Gaussian potential wells

$\begin{matrix}{{{V_{n}(x)} = {\sum\limits_{j = {- N}}^{N}{{- V_{0}}{\exp( {- \frac{( {x - {j\; L} - {n\frac{L}{3}}} )^{2}}{2\; \sigma^{2}}} )}}}},} & (71)\end{matrix}$

where n=0, 1, or 2, and N sets the extent of the landscape. Theprobability density ρ(x,t)dx for finding a Brownian particle within dxof position x at time t in state n evolves according to the masterequation,

ρ(y,t+T)=∫P _(n)(y,T|x,0)ρ(x,t)dx,  (72)

characterized by the propagator

P _(n)(y,T|x,0)=e ^(L) ^(n) ^((y)) Tδ(y−x),  (73)

where the Liouville operator for state n is

$\begin{matrix}{{{L_{n}(y)} = {D( {\frac{\partial^{2}}{\partial y^{2}} - {\beta \frac{\partial}{\partial y}{V_{n}^{\prime}(y)}}} )}},} & (74)\end{matrix}$

with

${{V_{n}^{\prime}(y)} = \frac{V_{n}}{y}},$

and where β⁻¹ is the thermal energy scale.

The master equation for a three-state cycle is

ρ(y,t+3T)=∫P ₁₂₃(y,3T|x,0)ρ(x,t)dx,  (75)

with the three-state propagator

P ₁₂₃(y,3T|x,0)=∫dy ₁ dy ₂ P ₃(y,T|y ₂,0)xP ₂(y ₂ ,T|y ₁,0)P ₁(y ₁,T|x,0)).  (76)

Because the landscape is periodic and analytic, Eq. (75) has asteady-state solution such that

ρ(x,t+3T)=ρ(x,t)  (77)

≡ρ₁₂₃(x).  (78)

The mean velocity of this steady-state then is given by

$\begin{matrix}{{v = {\int{{P_{123}( {y, {3\; T} \middle| x ,0} )}( \frac{y - x}{3\; T} ){\rho_{123}(x)}{x}\; {y}}}},} & (79)\end{matrix}$

where P₁₂₃(y,3T|x,0) is the probability for a particle originally atposition x to “jump” to position y by the end of one compete cycle,(y−x)/(3T) is the velocity associated with making such a jump, andρ₁₂₃(x) is the fraction of the available particles actually at x at thebeginning of the cycle in steady-state. This formulation is invariantwith respect to cyclic permutations of the states, so that the same fluxof particles would be measured at the end of each state. The averagevelocity v therefore describes the time-averaged flux of particlesdriven by the ratchet.

FIG. 12( a) shows numerical solutions of this system of equations forrepresentative values of the relative inter-well separation L/σ. If theinterval T between states is very short, particles are unable to keep upwith the evolving potential energy landscape, and so never travel farfrom their initial positions; the mean velocity vanishes in this limit.The transport speed v also vanishes as 1/T for large values of T becausethe induced drift becomes limited by the delay between states. If trapsin consecutive patterns are close enough (L=6.5σ in FIG. 12( a))particles jump forward at each transition with high probability,yielding a uniformly positive drift velocity. This transfer reaches itsmaximum efficiency for moderate cycle times, T/τ≈2√{square root over(2)}(L/σ)(βV₀)⁻¹. More widely separated traps (L=13σ in FIG. 12( a))yield more interesting behavior. Here, particles are able to keep upwith the forward-going wave for large values of T. Faster cycling,however, leads to flux reversal, characterized by negative values of v.

FIG. 12( a) further presents crossover from deterministic opticalperistalsis at L=6.5σ to thermal ratchet behavior with flux reversal atL=13σ for a three-state cycle of Gaussian well potentials at βV₀=8.5,σ=0.53 μm and D=0.33 μm²/sec. Intermediate curves are calculated forevenly spaced values of L. FIG. 12( b) presents an image of 20×5 arrayof holographic optical traps at L₀=6.7 μm FIG. 12( c) presents an imageof colloidal silica spheres 1.53 μm in diameter interacting with thearray. FIG. 12( d) presents the rate dependence of the induced driftvelocity for fixed inter-trap separation, L₀. FIG. 12( e) presents theseparation dependence for fixed inter-state delay, T=2 sec.

As an example of operation, this thermal ratchet protocol is implementedfor a sample of 1.53 μm diameter colloidal silica spheres (BangsLaboratories, lot number 5328) dispersed in water, using potentialenergy landscapes created from arrays of holographic optical traps. Thesample was enclosed in a hermetically sealed glass chamber roughly 40 μmthick created by bonding the edges of a coverslip to a microscope slideand was allowed to equilibrate to room temperature (21±1° C.) on thestage of a Zeiss S100TV Axiovert inverted optical microscope. A 100×NA1.4 oil immersion SPlan Apo objective lens was used to focus the opticaltweezer array into the sample and to image the spheres, whose motionswere captured with an NEC TI 324A low noise monochrome CCD camera. Themicrograph in FIG. 12( b) shows the focused light from a 20×5 array ofoptical traps formed by a phase hologram projected with a HamamatsuX7550 spatial light modulator. The tweezers are arranged in twenty-trapmanifolds 25 μm long separated by L₀=6.7 μm. Each trap is powered by anestimated 2.5±0.4 mW of laser light at 532 nm. The particles, whichappear in the bright-field micrograph in FIG. 12( c), are twice as denseas water and sediment to the lower glass surface, where they diffusefreely in the plane with a measured diffusion coefficient of D=0.33±0.03μm²/sec, which reflects the influence of the nearby wall. Out-of-planefluctuations were minimized by projecting the traps at the spheres'equilibrium height above the wall.

Three-state cycles of optical trapping patterns are projected in whichthe manifolds in FIG. 12( b) were displaced horizontally by −L₀/3, 0,and L₀/3, with inter-state delay times T ranging from 0.8 sec. to 10sec. The particles' motions were recorded as uncompressed digital videostreams for analysis. Between 40 and 60 particles were in the trappingpattern during a typical run, so that roughly 40 cycles sufficed toacquire reasonable statistics under each set of conditions withoutcomplications due to collisions. Particles outside the trapping patternare tracked to monitor their diffusion coefficients and to ensure theabsence of drifts in the supporting fluid. The results plotted in FIG.12( d) reveal flux reversal at T/τ≈0.03. Excellent agreement with Eq.(79) is obtained for βV₀=8.5±0.8 and σ=0.53±0.01 μm.

The appearance of flux reversal as one parameter is varied implies thatother parameters also should control the direction of motion. Indeed,flux reversal is obtained in FIG. 12( e) as the inter-trap separation isvaried from L=5.1 μm to 8.3 μm at fixed delay time, T=2 sec. Theseresults also agree well with predictions of Eq. (79), with no adjustableparameters. The same effect also should arise for different populationsin a heterogeneous sample with different values of D, V₀ and σ. In thiscase, distinct fractions can be induced to move simultaneously inopposite directions.

Such sensitivity of the transport direction to details of the dynamicsalso might play a role in the functioning of molecular motors such asmyosin-VI whose retrograde motion on actin filaments compared with othermyosins has excited much interest. This molecular motor is known to benonprocessive; its motion involves a diffusive search of the actinfilament's potential energy landscape, which nevertheless results inunidirectional hand-over-hand transport. These characteristics areconsistent with the present model's timing-based flux reversalmechanism, and could provide a basis to explain how small structuraldifferences among myosins could lead to oppositely directed transport.

E. Flux Reversal for Two-State Thermal Ratchets

Another exemplary embodiment of the present invention is presented fortwo state ratchets. A Brownian particle's random motions can berectified by a periodic potential energy landscape that alternatesbetween two states, even if both states are spatially symmetric. If thetwo states differ only by a discrete translation, the direction of theratchet-driven current can be reversed by changing their relativedurations. The present embodiment provides flux reversal in a symmetrictwo-state ratchet by tracking the motions of colloidal spheres movingthrough large arrays of discrete potential energy wells created withdynamic holographic optical tweezers. The model's simplicity and highdegree of symmetry suggest possible applications in molecular-scalemotors.

Until fairly recently, random thermal fluctuations were consideredimpediments to inducing motion in systems such as motors. Fluctuationscan be harnessed, however, through mechanisms such as stochasticresonance and thermal ratchets, as efficient transducers of input energyinto mechanical motion. Unlike conventional machines, which battlenoise, molecular-scale devices that exploit these processes actuallyrequite thermal fluctuations to operate.

The present embodiment creates thermal ratchets in which the randommotions of Brownian particles are rectified by a time-varying potentialenergy landscape. Even when the landscape has no overall slope and thusexerts no average force, directed motion still can result from theaccumulation of coordinated impulses. Most thermal ratchet models breakspatiotemporal symmetry by periodically translating, tilting orotherwise modulating a spatially asymmetric landscape. Inducing a fluxis almost inevitable in such systems unless they satisfy conditions ofspatiotemporal symmetry or supersymmetry. Even a spatially symmetriclandscape can induce a flux with appropriate driving. Unlikedeterministic motors, however, the direction of motion in these systemscan depend sensitively on implementation details.

A spatially symmetric three-state thermal ratchet is demonstrated formicrometer-scale colloidal particles implemented with arrays ofholographic optical tweezers, each of which constitutes a discretepotential energy well. Repeatedly displacing the array first by onethird of a lattice constant and then by two thirds breaks spatiotemporalsymmetry in a manner that induces a flux. Somewhat surprisingly, thedirection of motion depends sensitively on the duration of the statesrelative to the time required for a particle to diffuse the inter-trapseparation. The induced flux therefore can be canceled or even reversedby varying the rate of cycling, rather than the direction. This approachbuilds upon the pioneering demonstration of unidirectional flux inducedby a spatially asymmetric time-averaged optical ratchet, and ofreversible transitions driven by stochastic resonance in a dual-traprocking ratchet.

FIG. 13 presents a sequence of one complete cycle of aspatially-symmetric two-state ratchet potential comprised of discretepotential wells.

Here, flux induction and flux reversal is demonstrated in a symmetrictwo-state thermal ratchet implemented with dynamic holographic opticaltrap arrays. The transport mechanism for this two-state ratchet is moresubtle than the previous three-state model in that the direction ofmotion is not easily intuited from the protocol. Its capacity for fluxreversal in the absence of external loading, by contrast, can beinferred immediately by considerations of spatiotemporal symmetry. Thisalso differs from the three-state ratchet and the rocking double-tweezerin which flux reversal results from a finely tuned balance ofparameters.

FIG. 13 schematically therefore depicts how the two-state ratchetoperates. Each state consists of a pattern of discrete optical traps,modeled here as Gaussian wells of width σ and depth V₀, uniformlyseparated by a distance L>>σ. The first array of traps is extinguishedafter time T₁ and replaced immediately with a second array, which isdisplaced from the first by L/3. The second pattern is extinguishedafter time T₂ and replaced again by the first, thereby completing onecycle.

If the potential wells in the second state overlap those in the first,then trapped particles are handed back and forth between neighboringtraps as the states cycle, and no motion results. This also isqualitatively different from the three-state ratchet, whichdeterministically transfers particles forward under comparableconditions, in a process known as optical peristalsis. The only way thesymmetric two-state ratchet can induce motion is if trapped particlesare released when the states change and then diffuse freely.

FIG. 14( a) presents a displacement function ƒ(t) and FIG. 14( b)presents equivalent tilting-ratchet driving force, F(t)=−η{dot over(ƒ)}(t).

The motion of a Brownian particle in this system can be described withthe one-dimensional Langevin equation

η{dot over (x)}(t)=−V′(x(t)−ƒ(t))+ξ(t),  (80)

where η is the fluid's dynamic viscosity, V(x) is the potential energylandscape, V′(x)=∂V(x)/∂x is its derivative, and ξ(t) is adelta-correlated stochastic force representing thermal noise. Thepotential energy landscape in our system is spatially periodic withperiod L,

V(x+L)=V(x).  (81)

The time-varying displacement of the potential energy in our two-stateratchet is described by a periodic function ƒ(t) with period T=T₁+T₂,which is plotted in FIG. 14( a).

The equations describing this traveling potential ratchet can be recastinto the form of a tilting ratchet, which ordinarily would beimplemented by applying an oscillatory external force to objects on anotherwise fixed landscape. The appropriate coordinate transformation,y(t)=x(t)−ƒ(t), yields

η{dot over (y)}(t)=−V′(y(t))+F(t)+ξ(t),  (82)

where F(t)=−η{dot over (ƒ)}(t) is the effective driving force. Becauseƒ(t) has a vanishing mean, the average velocity of the original problemis the same as that of the transformed tilting ratchet

{dot over (x)}

=

{dot over (y)}

, where the angle brackets imply both an ensemble average and an averageover a period T.

Reimann has demonstrated that a steady-state flux,

{dot over (y)}

≠0, develops in any tilting ratchet that breaks both spatiotemporalsymmetry,

V(y)=V(−y), and −F(t)=F(t+T/2),  (83)

and also spatiotemporal supersymmetry,

−V(y)=V(y+L/2), and −F(t)=F(−t).  (84)

for any Δt. No flux results if either of Eqs. (83) or (84) is satisfied.

The optical trapping potential depicted in FIG. 13 is symmetric but notsupersymmetric. Provided that F(t) violates the symmetry condition inEq. (83), the ratchet must induce directed motion. Although F(t) issupersymmetric, as can be seen in FIG. 14( b), it is symmetric only whenT₁=T₂. Consequently, we expect a particle current for T₁≠T₂. The zerocrossing at T₁=T₂ furthermore portends flux reversal on either side ofthe equality.

FIG. 15 presents steady-state drift velocity as a function of therelative dwell time, T₂/T₁, for βV₀=3.04, L=5.2 μm, σ=0.80 μm, andvarious values of T/τ. Transport is optimized under these conditions byrunning the ratchet at T/τ=0.193.

The steady-state velocity is calculated for this system by solving themaster equation associated with Eq. (80). The probability for a drivenBrownian particle to drift from position x₀ to within dx of position xduring the interval t, is given by the propagator

P(x,t|x ₀,0)dx=e ^(∫) ^(t) ^(L(x,t′)dt′)δ(x−x ₀)dx,  (85)

where the Liouville operator is

$\begin{matrix}{{{L( {x,t} )} = {D( {\frac{\partial^{2}}{\partial x^{2}} + {\beta \frac{\partial}{\partial x}{V^{\prime}( {x,t} )}}} )}},} & (86)\end{matrix}$

and where β⁻¹ is the thermal energy scale. The steady-state particledistribution ρ(x) is an eigenstate of the master equation

ρ(x)=∫P(x,t|x ₀)ρ(x ₀)dx ₀,  (87)

and the associated steady-state flux is

$\begin{matrix}{v = {\int{\frac{x - x_{0}}{T}{\rho ( x_{0} )}{P( {x, T \middle| x_{0} ,0} )}{x}{{x_{0}}.}}}} & (88)\end{matrix}$

The natural length scale in this problem is L, the inter-trap spacing ineither state. The natural time scale, τ=L²/(2D), is the time requiredfor particles of diffusion constant D to diffuse this distance.

FIG. 15 shows how v varies with T₂/T₁ for various values of T/τ forexperimentally accessible values of V₀, σ, and L. As anticipated, thenet drift vanishes for T₂=T₁. Less obviously, the induced flux isdirected from each well in the longer-duration state toward the nearestwell in the short-lived state. The flux falls off as 1/T in the limit oflarge T because the particles spend increasingly much of their timelocalized in traps. It also diminishes for short T because the particlescannot keep up with the landscape's evolution. In between, the range offluxes can be tuned with T.

FIG. 16( a) presents an image of 5×20 array of holographic optical trapsat L=5.2 μm FIG. 16( b) presents a video micrograph of colloidal silicaspheres 1.53 μm in diameter trapped in the middle row of the array atthe start of an experimental run. FIGS. 16( c) and 16(d) present thetime evolution of the measured probability density for finding particlesat T₂=0.8 see and T₂=8.6 sec, respectively, with T₁ fixed at 3 sec. FIG.16( e) presents the time evolution of the particles' mean positioncalculated from the distribution functions in 16(c) and 16(d). Theslopes of linear fits provide estimates for the induced drift velocity,which can be compared with displacements calculated with Eq. (89) forβV₀=2.75, and σ=0.65 μm FIG. 16( f) presents the measured drift speed asa function of relative dwell time T₂/T₁, compared with predictions ofEq. (88).

As an example, this method is implemented for a sample of 1.53 μmdiameter colloidal silica spheres (Bangs Laboratories, lot number 5328)dispersed in water, using potential energy landscapes created fromarrays of holographic optical traps. The sample was enclosed in ahermetically sealed glass chamber roughly 40 μm thick created by bondingthe edges of a coverslip to a microscope slide, and was allowed toequilibrate to room temperature (21±1° C.) on the stage of a Zeiss S1002TV Axiovert inverted optical microscope. A 100×NA 1.4 oil immersionSPlan Apo objective lens was used to focus the optical tweezer arrayinto the sample and to image the spheres, whose motions were capturedwith an NEC TI 324A low noise monochrome CCD camera. The micrograph inFIG. 16( a) shows the focused light from a 5×20 array of optical trapsformed by a phase hologram projected with a Hamamatsu X7550 spatiallight modulator [17]. The tweezers are arranged in twenty-trap manifolds37 μm long separated by L=5.2 μm. Each trap is powered by an estimated2.5±0.4 mW of laser light at 532 nm. The particles, which appear in thebright-field micrograph in FIG. 16( b), are twice as dense as water andsediment to the lower glass surface, where they diffuse freely in theplane with a measured diffusion coefficient of D=0.33±0.03 μm²/sec. Thisestablishes the characteristic time scale for the system of τ=39.4 sec,which is quite reasonable for digital video microscopy studies.Out-of-plane fluctuations were minimized by focusing the traps at thespheres' equilibrium height above the wall.

Two-state cycles of optical trapping patterns are projected in which themanifolds in FIG. 16( a) were alternately displaced in the spheres'equilibrium plane by L/3, with the duration of the first state fixed atT₁=3 sec and T₂ ranging from 0.8 sec to 14.7 sec. To measure the fluxinduced by this cycling potential energy landscape for one value of T₂,we first gathered roughly two dozen particles in the middle row of trapsin state 1, as shown in FIG. 16( b), and then projected up to onehundred periods of two-state cycles. The particles' motions wererecorded as uncompressed digital video streams for analysis. Theirtime-resolved trajectories then were averaged over the transversedirection into the probability density, ρ(x,t)Δx, for finding particleswithin Δx=0.13 μm of position x after time t. We also tracked particlesoutside the trapping pattern to monitor their diffusion coefficients andto ensure the absence of drifts in the supporting fluid. Starting fromthis well-controlled initial condition resolves any uncertaintiesarising from the evolution of nominally random initial conditions.

FIGS. 16( c) and 16(d) show the spatially-resolved time evolution ofρ(x,t) for T₂=0.8 see <T₁ and T₂=8.6 see >T₁ In both cases, theparticles spend most of their time localized in traps, visible here asbright stripes, occasionally using the shorter-lived traps asspringboards to neighboring wells in the longer-lived state. The meanparticle position

x(t)

=xρ(x,t)dx advances as the particles make these jumps, with theassociated results plotted in FIG. 16( e).

The speed with which an initially localized state, ρ(x,0)≈δ(x), advancesdiffers from the steady-state speed plotted, in FIG. 15, but still canbe calculated as the first moment of the propagator,

x(t)

=∫yP(y,t|0,0)dy.  (89)

Numerical analysis reveals a nearly constant mean speed that agreesquite closely with the steady-state speed from Eq. (88).

Fitting traces such as those in FIG. 16( e) to linear trends providesestimates for the ratchet-induced flux, which are plotted in FIG. 16(f). The solid curve in FIG. 16( f) shows excellent agreement withpredictions of Eq. (89) for βV₀=2.75±0.5 and σ=0.65±0.05 μm.

The two-state ratchet method presented herein therefore involvesupdating the optical intensity pattern to translate the physicallandscape. However, the same principles can be applied to systems inwhich the landscape remains fixed and the object undergoes cyclictransitions between two states. FIG. 17 depicts a model for an activetwo-state walker on a fixed physical landscape that is inspired by thebiologically relevant transport of single myosin head groups along actinfilaments. The walker consists of a head group that interacts withlocalized potential energy wells periodically distributed on thelandscape. It also is attached to a lever arm that uses an externalenergy source to translate the head group by a distance somewhat smallerthan the inter-well separation. The other end of the lever arm isconnected to the payload, whose viscous drag would provide the leveragenecessary to translate the head group between the extended and retractedstates. Switching between the walker's two states is equivalent to thetwo-state translation of the potential energy landscape in ourexperiments, and thus would have the effect of translating the walker inthe direction of the shorter-lived state. A similar model in which atwo-state walker traverses a spatially asymmetric potential energylandscape yields deterministic motion at higher efficiency than thepresent model. It does not, however, allow for reversibility. The lengthof the lever arm and the diffusivity of the motor's body and payloaddetermine the ratio T/τ and thus the motor's efficiency. The two-stateratchet's direction does not depend on T/τ, however, even under heavyloading. This differs from the three-state ratchet, in which T/τ alsocontrols the direction of motion. This protocol could be used in thedesign of mesoscopic motors based on synthetic macromolecules ormicroelectromechanical systems (MEMS).

The foregoing description of embodiments of the present invention havebeen presented for purposes of illustration and description. It is notintended to be exhaustive or to limit the present invention to theprecise form disclosed, and modifications and variations are possible inlight of the above teachings or may be acquired from practice of thepresent invention. The embodiments were chosen and described in order toexplain the principles of the present invention and its practicalapplication to enable one skilled in the art to utilize the presentinvention in various embodiments and with various modifications as aresuited to the particular use contemplated.

1. A method for manipulation of a plurality of objects comprising thesteps of: providing a shaping source; applying the shaping source tocreate a spatially symmetric potential energy landscape; applying thepotential energy landscape to a plurality of objects, thereby trappingat least a portion of the plurality of objects in the potential energylandscape; and spatially moving the potential energy landscape tomanipulate the plurality of objects.
 2. The method as defined in claim 1wherein the shaping source is selected from the group of an opticalsource, a hologram, an electrical source, and a textured surface.
 3. Themethod as defined in claim 1 wherein the potential energy landscapecomprises a pattern of potential wells.
 4. The method as defined inclaim 1 wherein the shaping source comprises an optical holographicsource.
 5. The method as defined in claim 4 wherein the potential energylandscape comprises a pattern of optical potential wells.
 6. The methodas defined in claim 1 wherein the step of moving the potential energylandscape includes dynamically changing the shaping source.
 7. Themethod as defined in claim 1 wherein the potential energy landscapecomprises an optical trap array.
 8. The method as defined in claim 1wherein the plurality of objects undergo thermal motion when thepotential energy landscape is extinguished, thereby allowing theplurality of objects to diffuse freely.
 9. The method as defined inclaim 1 wherein the potential energy landscape includes at least threedifferent ones of the landscape projected over time and space tomanipulate the plurality of objects.
 10. The method as defined in claim1 wherein the potential energy landscape is projected in threedimensions and the objects being trapped have at least molecular sizedimensions.
 11. The method as defined in claim 1 wherein the potentialenergy landscape comprises at least one of a movable arm and a polymerarm adjustable in space.
 12. The method as defined in claim 11 whereinthe at least one of movable arm and polymer arm carries out a step of atleast one of sorting and pumping the objects.
 13. The method as definedin claim 1 wherein the potential energy landscape comprises at least oneof a radial ratchet and a linear ratchet.
 14. A method of optimizing aplurality of optical traps, comprising: providing a beam of light from alight source; imprinting the collimated beam of light withcomputer-generated holography; propagating a plurality of independentbeams of light from the collimated beam of light through a diffractiveoptical element; relaying the independent beams to an input aperture ofa lens; focusing the independent beams on an objective focal plane; andfocusing an undiffracted portion of the beam of light, wherein the beamof light converges as it passes through the diffractive optical element,causing the undiffracted portion of beam of light to focus upstream ofthe objective focal plane.
 15. The method of claim 1, wherein thecomputer-generated holography includes a wavefront-shaping phasefunction.
 16. The method of claim 1, further comprising including a beamblock in an intermediate focal plane, the beam block spatially filteringthe undiffracted portion of the beam of light.
 17. The method of claim1, wherein the beam of light comprises a laser beam.